Banach Journal of Mathematical Analysis Articles (Project Euclid)
http://projecteuclid.org/euclid.bjma
The latest articles from Banach Journal of Mathematical Analysis on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTMon, 07 Feb 2011 17:20 ESThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
http://projecteuclid.org/
Note on extreme points in Marcinkiewicz function spaces
http://projecteuclid.org/euclid.bjma/1272374667
<strong> Anna Kaminska </strong>, <strong> Anca M. Parrish </strong><p><strong>Source: </strong>Banach J. Math. Anal., Volume 4, Number 1, 1--12.</p><p><strong>Abstract:</strong><br/>
We show that the unit ball of the subspace $M_W^0$ of ordered continuous
elements of $M_W$ has no extreme points, where $M_W$ is the Marcinkiewicz
function space generated by a decreasing weight function $w$ over the interval
$(0,\infty)$ and $W(t) = \int_0^tw$, $t\in(0,\infty)$. We also present here a
proof of the fact that a function $f$ in the unit ball of $M_W$ is an extreme
point if and only if $f^*=w$.
</p>projecteuclid.org/euclid.bjma/1272374667_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTDisjointness-preserving orthogonally additive operators in vector latticeshttps://projecteuclid.org/euclid.bjma/1529114495<strong>Nariman Abasov</strong>, <strong>Marat Pliev</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 3, 730--750.</p><p><strong>Abstract:</strong><br/>
In this article, we investigate disjointness-preserving orthogonally additive operators in the setting of vector lattices. First, we present a formula for the band projection onto the band generated by a single positive, disjointness-preserving, order-bounded, orthogonally additive operator. Then we prove a Radon–Nikodým theorem for a positive, disjointness-preserving, order-bounded, orthogonally additive operator defined on a vector lattice $E$ , taking values in a Dedekind-complete vector lattice $F$ . We conclude by obtaining an analytical representation for a nonlinear lattice homomorphism between order ideals of spaces of measurable almost everywhere finite functions.
</p>projecteuclid.org/euclid.bjma/1529114495_20180703040033Tue, 03 Jul 2018 04:00 EDTReflexive sets of operatorshttps://projecteuclid.org/euclid.bjma/1526630423<strong>Janko Bračič</strong>, <strong>Cristina Diogo</strong>, <strong>Michal Zajac</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 3, 751--771.</p><p><strong>Abstract:</strong><br/>
For a set $\mathcal{M}$ of operators on a complex Banach space $\mathscr{X}$ , the reflexive cover of $\mathcal{M}$ is the set $\operatorname{Ref}(\mathcal{M})$ of all those operators $T$ satisfying $Tx\in\overline{\mathcal{M}x}$ for every $x\in\mathscr{X}$ . Set $\mathcal{M}$ is reflexive if $\operatorname{Ref}(\mathcal{M})=\mathcal{M}$ . The notion is well known, especially for Banach algebras or closed spaces of operators, because it is related to the problem of invariant subspaces. We study reflexivity for general sets of operators. We are interested in how the reflexive cover behaves towards basic operations between sets of operators. It is easily seen that the intersection of an arbitrary family of reflexive sets is reflexive, as well. However this does not hold for unions, since the union of two reflexive sets of operators is not necessarily a reflexive set. We give some sufficient conditions under which the union of reflexive sets is reflexive. We explore how the reflexive cover of the sum (resp., the product) of two sets is related to the reflexive covers of summands (resp., factors). We also study the relation between reflexivity and convexity, with special interest in the question: under which conditions is the convex hull of a reflexive set reflexive?
</p>projecteuclid.org/euclid.bjma/1526630423_20180703040033Tue, 03 Jul 2018 04:00 EDTPartial actions of $C^{*}$ -quantum groupshttps://projecteuclid.org/euclid.bjma/1517281422<strong>Franziska Kraken</strong>, <strong>Paula Quast</strong>, <strong>Thomas Timmermann</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 4, 843--872.</p><p><strong>Abstract:</strong><br/>
Partial actions of groups on $C^{*}$ -algebras and the closely related actions and coactions of Hopf algebras have received much attention in recent decades. They arise naturally as restrictions of their global counterparts to noninvariant subalgebras, and the ambient enveloping global (co)actions have proven useful for the study of associated crossed products. In this article, we introduce the partial coactions of $C^{*}$ -bialgebras, focusing on $C^{*}$ -quantum groups, and we prove the existence of an enveloping global coaction under mild technical assumptions. We also show that partial coactions of the function algebra of a discrete group correspond to partial actions on direct summands of a $C^{*}$ -algebra, and we relate partial coactions of a compact or its dual discrete $C^{*}$ -quantum group to partial coactions or partial actions of the dense Hopf subalgebra. As a fundamental example, we associate to every discrete $C^{*}$ -quantum group a quantum Bernoulli shift.
</p>projecteuclid.org/euclid.bjma/1517281422_20180927040105Thu, 27 Sep 2018 04:01 EDTOn domains of unbounded derivations of generalized B $^{*}$ -algebrashttps://projecteuclid.org/euclid.bjma/1524211222<strong>Martin Weigt</strong>, <strong>Ioannis Zarakas</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 4, 873--908.</p><p><strong>Abstract:</strong><br/>
We study properties under which the domain of a closed derivation $\delta:D(\delta)\rightarrow A$ of a generalized B $^{*}$ -algebra $A$ remains invariant under analytic functional calculus. For a complete, generalized B $^{*}$ -algebra with jointly continuous multiplication, two sufficient conditions are assumed: that the unit of $A$ belongs to the domain of the derivation, along with a condition related to the coincidence $\sigma_{A}(x)=\sigma_{D(\delta)}(x)$ of the (Allan) spectra for every element $x\in D(\delta)$ . Certain results are derived concerning the spectra for a general element of the domain, in the realm of a domain which is advertibly complete or enjoys the Q-property. For a closed $*$ -derivation $\delta$ of a complete GB $^{*}$ -algebra with jointly continuous multiplication such that $1\in D(\delta)$ and $x$ a normal element of the domain, $f(x)\in D(\delta)$ for every analytic function on a neighborhood of the spectrum of $x$ . We also give an example of a closed derivation of a GB $^{*}$ -algebra which does not contain the identity element. A condition for a closed derivation of a GB $^{*}$ -algebra $A$ to be the generator of a one-parameter group of automorphisms of $A$ is provided along with a generalization of the Lumer–Phillips theorem for complete locally convex spaces.
</p>projecteuclid.org/euclid.bjma/1524211222_20180927040105Thu, 27 Sep 2018 04:01 EDTBoundedness of Hausdorff operators on Hardy spaces in the Heisenberg grouphttps://projecteuclid.org/euclid.bjma/1529632824<strong>Qingyan Wu</strong>, <strong>Zunwei Fu</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 4, 909--934.</p><p><strong>Abstract:</strong><br/>
In the setting of the Heisenberg group, we define weighted Hardy spaces by means of their atomic characterization, and we establish the (sharp) boundedness of Hausdorff operators on power-weighted Hardy spaces. Moreover, we obtain sufficient and necessary conditions for the boundedness of Hausdorff operators on local Hardy spaces in the Heisenberg group.
</p>projecteuclid.org/euclid.bjma/1529632824_20180927040105Thu, 27 Sep 2018 04:01 EDTOn approximation properties of $l_{1}$ -type spaceshttps://projecteuclid.org/euclid.bjma/1531209675<strong>Maciej Ciesielski</strong>, <strong>Grzegorz Lewicki</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 4, 935--954.</p><p><strong>Abstract:</strong><br/>
Let $(X_{n}\Vert \cdot \Vert _{n})$ denote a sequence of real Banach spaces. Let
\[X=\bigoplus_{1}X_{n}=\{(x_{n}):x_{n}\in X_{n}\hbox{ for any }n\in \mathbb{N},\sum_{n=1}^{\infty}\Vert x_{n}\Vert _{n}\lt \infty\}.\] In this article, we investigate some properties of best approximation operators associated with finite-dimensional subspaces of $X$ . In particular, under a number of additional assumptions on $(X_{n})$ , we characterize finite-dimensional Chebyshev subspaces $Y$ of $X$ . Likewise, we show that the set
\[\mathrm{Nuniq}=\{x\in X:\operatorname{card}(P_{Y}(x))\gt 1\}\] is nowhere dense in $Y$ , where $P_{Y}$ denotes the best approximation operator onto $Y$ . Finally, we demonstrate various (mainly negative) results on the existence of continuous selection for metric projection and we provide examples illustrating possible applications of our results.
</p>projecteuclid.org/euclid.bjma/1531209675_20180927040105Thu, 27 Sep 2018 04:01 EDTInterpolating inequalities for functions of positive semidefinite matriceshttps://projecteuclid.org/euclid.bjma/1531209674<strong>Ahmad Al-Natoor</strong>, <strong>Omar Hirzallah</strong>, <strong>Fuad Kittaneh</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 4, 955--969.</p><p><strong>Abstract:</strong><br/>
Let $A$ , $B$ be positive semidefinite $n\times n$ matrices, and let $\alpha\in(0,1)$ . We show that if $f$ is an increasing submultiplicative function on $[0,\infty)$ with $f(0)=0$ such that $f(t)$ and $f^{2}(t^{1/2})$ are convex, then \begin{eqnarray*}\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert f(\llvert AB\rrvert )\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert^{2}&\leq&f^{4}(\frac{1}{(4\alpha(1-\alpha))^{1/4}})(\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert(\alpha f(A)+(1-\alpha )f(B))^{2}\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert\\&&{}\times \big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert((1-\alpha)f(A)+\alpha f(B))^{2}\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert)\end{eqnarray*} for every unitarily invariant norm. Moreover, if $\alpha\in{}[0,1]$ and $X$ is an $n\times n$ matrix with $X\neq0$ , then \begin{eqnarray*}&&\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert f(\llvert AXB\rrvert )\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert^{2}\\&&\quad \leq\frac{f(\Vert X\Vert)}{\Vert X\Vert}\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert\alpha f^{2}(A)X+(1-\alpha)Xf^{2}(B)\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert \big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert(1-\alpha)f^{2}(A)X+\alpha Xf^{2}(B)\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert\end{eqnarray*} for every unitarily invariant norm. These inequalities present generalizations of recent results of Zou and Jiang and of Audenaert.
</p>projecteuclid.org/euclid.bjma/1531209674_20180927040105Thu, 27 Sep 2018 04:01 EDTHigher-order compact embeddings of function spaces on Carnot–Carathéodory spaceshttps://projecteuclid.org/euclid.bjma/1535594467<strong>Martin Franců</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 4, 970--994.</p><p><strong>Abstract:</strong><br/>
A sufficient condition for higher-order compact embeddings on bounded domains in Carnot–Carathéodory spaces is established for the class of rearrangement-invariant function spaces. The condition is expressed in terms of compactness of a suitable $1$ -dimensional integral operator depending on the isoperimetric function relative to the Carnot–Carathéodory structure of the relevant sets. The general result is then applied to particular Sobolev spaces built upon Lebesgue and Lorentz spaces.
</p>projecteuclid.org/euclid.bjma/1535594467_20180927040105Thu, 27 Sep 2018 04:01 EDTPhillips symmetric operators and their extensionshttps://projecteuclid.org/euclid.bjma/1536048015<strong>Sergii Kuzhel</strong>, <strong>Leonid Nizhnik</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 4, 995--1016.</p><p><strong>Abstract:</strong><br/>
This article is devoted to the investigation of self-adjoint (and, more generally, proper) extensions of Phillips symmetric operators (PSO). A closed densely defined symmetric operator with equal defect numbers is considered a Phillips symmetric operator if its characteristic function is a constant on $\mathbb{C}_{+}$ . We present equivalent definitions of PSO and prove that proper extensions with real spectra of a given PSO are similar to each other. Our results imply that one-point interaction of the momentum operator $i\frac{d}{dx}+\alpha\delta(x-y)$ leads to unitarily equivalent self-adjoint operators with Lebesgue spectra. Self-adjoint operators with nontrivial spectral properties can be obtained as a result of more complicated perturbations of the momentum operator. In this way, we study special classes of perturbations which can be characterized as one-point interactions defined by the nonlocal potential $\gamma\in{L_{2}(\mathbb{R})}$ .
</p>projecteuclid.org/euclid.bjma/1536048015_20180927040105Thu, 27 Sep 2018 04:01 EDTWavelet characterizations of Musielak–Orlicz Hardy spaceshttps://projecteuclid.org/euclid.bjma/1536048016<strong>Xing Fu</strong>, <strong>Dachun Yang</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 4, 1017--1046.</p><p><strong>Abstract:</strong><br/>
In this article, via establishing a new atomic characterization of the Musielak–Orlicz Hardy space $H^{\varphi}({\mathbb{R}}^{n})$ [which is essentially deduced from the known molecular characterization of $H^{\varphi}({\mathbb{R}}^{n})$ ] and some estimates on a new discrete Littlewood–Paley $g$ -function and a Peetre-type maximal function, together with using the known intrinsic $g$ -function characterization of $H^{\varphi}({\mathbb{R}}^{n})$ , the authors obtain several equivalent characterizations of $H^{\varphi}({\mathbb{R}}^{n})$ in terms of wavelets, which extend the wavelet characterizations of both Orlicz–Hardy spaces and the weighted Hardy spaces, and are available to the typical and useful Musielak–Orlicz Hardy space $H^{\log}({\mathbb{R}}^{n})$ . The novelty of this approach is that the new adapted atomic characterization of $H^{\varphi}({\mathbb{R}}^{n})$ compensates the inconvenience in applications of the supremum appearing in the original definition of atoms, which play crucial roles in the proof of the main theorem of this article and may have further potential applications.
</p>projecteuclid.org/euclid.bjma/1536048016_20180927040105Thu, 27 Sep 2018 04:01 EDTBimonotone maps on semiprime Banach algebrashttps://projecteuclid.org/euclid.bjma/1536653148<strong>M. Burgos</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 4, 1047--1063.</p><p><strong>Abstract:</strong><br/>
In this article, we investigate the properties of the sharp partial order in unital rings, and we study additive maps preserving the minus partial order in both directions in the setting of unital semiprime Banach algebras with essential socle.
</p>projecteuclid.org/euclid.bjma/1536653148_20180927040105Thu, 27 Sep 2018 04:01 EDTWeak boundedness of operator-valued Bochner–Riesz means for the Dunkl transformhttps://projecteuclid.org/euclid.bjma/1536653147<strong>Maofa Wang</strong>, <strong>Bang Xu</strong>, <strong>Jian Hu</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 4, 1064--1083.</p><p><strong>Abstract:</strong><br/>
We consider operator-valued Bochner–Riesz means with weight function $h_{\kappa}^{2}$ under a finite reflection group for the Dunkl transform. We establish the maximal inequality of the weighted Hardy–Littlewood maximal function, and we apply it to the maximal inequality of operator-valued Bochner–Riesz means $B^{\delta}_{R}(h^{2}_{\kappa};f)(x)$ for $\delta\gt \lambda_{\kappa}:=\frac{d-1}{2}+\sum_{j=1}^{d}\kappa_{j}$ . Furthermore, we also obtain the corresponding pointwise convergence theorem.
</p>projecteuclid.org/euclid.bjma/1536653147_20180927040105Thu, 27 Sep 2018 04:01 EDTRegularities of semigroups, Carleson measures and the characterizations of BMO-type spaces associated with generalized Schrödinger operatorshttps://projecteuclid.org/euclid.bjma/1539590538<strong>Yuanyuan Hao</strong>, <strong>Pengtao Li</strong>, <strong>Kai Zhao</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 1--25.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{L}=-\Delta+\mu$ be the generalized Schrödinger operator on $\mathbb{R}^{n},n\geq3$ , where $\Delta$ is the Laplacian and $\mu\notequiv0$ is a nonnegative Radon measure on $\mathbb{R}^{n}$ . In this article, we introduce two families of Carleson measures $\{d\nu_{h,k}\}$ and $\{d\nu_{P,k}\}$ generated by the heat semigroup $\{e^{-t\mathcal{L}}\}$ and the Poisson semigroup $\{e^{-t\sqrt{\mathcal{L}}}\}$ , respectively. By the regularities of semigroups, we establish the Carleson measure characterizations of BMO-type spaces $\mathrm{BMO}_{\mathcal{L}}(\mathbb{R}^{n})$ associated with the generalized Schrödinger operators.
</p>projecteuclid.org/euclid.bjma/1539590538_20181218040349Tue, 18 Dec 2018 04:03 ESTContinuous generalization of Clarkson–McCarthy inequalitieshttps://projecteuclid.org/euclid.bjma/1538121808<strong>Dragoljub J. Kečkić</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 26--46.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a compact Abelian group, let $\mu$ be the corresponding Haar measure, and let $\hat{G}$ be the Pontryagin dual of $G$ . Furthermore, let $\mathcal{C}_{p}$ denote the Schatten class of operators on some separable infinite-dimensional Hilbert space, and let $L^{p}(G;\mathcal{C}_{p})$ denote the corresponding Bochner space. If $G\ni\theta\mapsto A_{\theta}$ is the mapping belonging to $L^{p}(G;\mathcal{C}_{p})$ , then \begin{eqnarray*}\sum_{k\in\hat{G}}\Vert \int_{G}\overline{k(\theta)}A_{\theta}\,\mathrm{d}\theta\Vert _{p}^{p}&\le&\int_{G}\Vert A_{\theta}\Vert _{p}^{p}\,\mathrm{d}\theta,\quad p\ge2,\\\sum_{k\in\hat{G}}\Vert \int_{G}\overline{k(\theta)}A_{\theta}\,\mathrm{d}\theta\Vert _{p}^{p}&\le&(\int_{G}\Vert A_{\theta}\Vert _{p}^{q}\,\mathrm{d}\theta )^{p/q},\quad p\ge2,\\\sum_{k\in\hat{G}}\Vert \int_{G}\overline{k(\theta)}A_{\theta}\,\mathrm{d}\theta\Vert _{p}^{q}&\le&(\int_{G}\Vert A_{\theta}\Vert _{p}^{p}\,\mathrm{d}\theta )^{q/p},\quad p\le2.\end{eqnarray*} If $G$ is a finite group, then the previous equations comprise several generalizations of Clarkson–McCarthy inequalities obtained earlier (e.g., $G=\mathbf{Z}_{n}$ or $G=\mathbf{Z}_{2}^{n}$ ), as well as the original inequalities, for $G=\mathbf{Z}_{2}$ . We also obtain other related inequalities.
</p>projecteuclid.org/euclid.bjma/1538121808_20181218040349Tue, 18 Dec 2018 04:03 ESTParametric Marcinkiewicz integrals with rough kernels acting on weak Musielak–Orlicz Hardy spaceshttps://projecteuclid.org/euclid.bjma/1540865070<strong>Bo Li</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 47--63.</p><p><strong>Abstract:</strong><br/>
Let $\varphi:\mathbb{R}^{n}\times[0,\infty)\to[0,\infty)$ satisfy that $\varphi(x,\cdot)$ , for any given $x\in\mathbb{R}^{n}$ , is an Orlicz function and that $\varphi(\cdot ,t)$ is a Muckenhoupt $A_{\infty}$ weight uniformly in $t\in(0,\infty)$ . The weak Musielak–Orlicz Hardy space $\mathit{WH}^{\varphi}(\mathbb{R}^{n})$ is defined to be the set of all tempered distributions such that their grand maximal functions belong to the weak Musielak–Orlicz space $\mathit{WL}^{\varphi}(\mathbb{R}^{n})$ . For parameter $\rho\in(0,\infty)$ and measurable function $f$ on $\mathbb{R}^{n}$ , the parametric Marcinkiewicz integral $\mu _{\Omega}^{\rho}$ related to the Littlewood–Paley $g$ -function is defined by setting, for all $x\in\mathbb{R}^{n}$ ,
\[\mu^{\rho}_{\Omega}(f)(x):=(\int_{0}^{\infty}\vert\int_{|x-y|\leq t}\frac{\Omega(x-y)}{|x-y|^{n-\rho}}f(y){dy}\vert^{2}\frac{dt}{t^{2\rho+1}})^{1/2},\] where $\Omega$ is homogeneous of degree zero satisfying the cancellation condition.
In this article, we discuss the boundedness of the parametric Marcinkiewicz integral $\mu_{\Omega}^{\rho}$ with rough kernel from weak Musielak–Orlicz Hardy space $\mathit{WH}^{\varphi}(\mathbb{R}^{n})$ to weak Musielak–Orlicz space $\mathit{WL}^{\varphi}(\mathbb{R}^{n})$ . These results are new even for the classical weighted weak Hardy space of Quek and Yang, and probably new for the classical weak Hardy space of Fefferman and Soria.
</p>projecteuclid.org/euclid.bjma/1540865070_20181218040349Tue, 18 Dec 2018 04:03 ESTWeyl-almost periodic solutions and asymptotically Weyl-almost periodic solutions of abstract Volterra integro-differential equationshttps://projecteuclid.org/euclid.bjma/1540454496<strong>Marko Kostić</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 64--90.</p><p><strong>Abstract:</strong><br/>
The main purpose of this article is to investigate Weyl-almost periodic solutions and asymptotically Weyl-almost periodic solutions of abstract Volterra integro-differential equations and inclusions. The class of asymptotically Weyl-almost periodic functions that we introduce here seems not to have been considered elsewhere, even in the scalar-valued case. We analyze the Weyl-almost periodic and asymptotically Weyl-almost periodic properties of convolution products and various types of degenerate solution operator families subgenerated by multivalued linear operators.
</p>projecteuclid.org/euclid.bjma/1540454496_20181218040349Tue, 18 Dec 2018 04:03 ESTOn the unit sphere of positive operatorshttps://projecteuclid.org/euclid.bjma/1540865071<strong>Antonio M. Peralta</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 91--112.</p><p><strong>Abstract:</strong><br/>
Given a $C^{*}$ -algebra $A$ , let $S(A^{+})$ denote the set of positive elements in the unit sphere of $A$ . Let $H_{1}$ , $H_{2}$ , $H_{3}$ , and $H_{4}$ be complex Hilbert spaces, where $H_{3}$ and $H_{4}$ are infinite-dimensional and separable. In this article, we prove a variant of Tingley’s problem by showing that every surjective isometry $\Delta:S(B(H_{1})^{+})\to S(B(H_{2})^{+})$ (resp., $\Delta:S(K(H_{3})^{+})\to S(K(H_{4})^{+})$ ) admits a unique extension to a surjective complex linear isometry from $B(H_{1})$ onto $B(H_{2})$ (resp., from $K(H_{3})$ onto $K(H_{4})$ ). This provides a positive answer to a conjecture recently posed by Nagy.
</p>projecteuclid.org/euclid.bjma/1540865071_20181218040349Tue, 18 Dec 2018 04:03 ESTAnalytic aspects of evolution algebrashttps://projecteuclid.org/euclid.bjma/1543395629<strong>P. Mellon</strong>, <strong>M. Victoria Velasco</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 113--132.</p><p><strong>Abstract:</strong><br/>
We prove that every evolution algebra $A$ is a normed algebra, for an $l_{1}$ -norm defined in terms of a fixed natural basis. We further show that a normed evolution algebra $A$ is a Banach algebra if and only if $A=A_{1}\oplus A_{0}$ , where $A_{1}$ is finite-dimensional and $A_{0}$ is a zero-product algebra. In particular, every nondegenerate Banach evolution algebra must be finite-dimensional and the completion of a normed evolution algebra is therefore not, in general, an evolution algebra. We establish a sufficient condition for continuity of the evolution operator $L_{B}$ of $A$ with respect to a natural basis $B$ , and we show that $L_{B}$ need not be continuous. Moreover, if $A$ is finite-dimensional and $B=\{e_{1},\ldots,e_{n}\}$ , then $L_{B}$ is given by $L_{e}$ , where $e=\sum_{i}e_{i}$ and $L_{a}$ is the multiplication operator $L_{a}(b)=ab$ , for $b\in A$ . We establish necessary and sufficient conditions for convergence of $(L_{a}^{n}(b))_{n}$ , for all $b\in A$ , in terms of the multiplicative spectrum $\sigma_{m}(a)$ of $a$ . Namely, $(L_{a}^{n}(b))_{n}$ converges, for all $b\in A$ , if and only if $\sigma_{m}(a)\subseteq\Delta\cup\{1\}$ and $\nu(1,a)\leq1$ , where $\nu(1,a)$ denotes the index of $1$ in the spectrum of $L_{a}$ .
</p>projecteuclid.org/euclid.bjma/1543395629_20181218040349Tue, 18 Dec 2018 04:03 ESTQuantitative weighted bounds for the composition of Calderón–Zygmund operatorshttps://projecteuclid.org/euclid.bjma/1540454497<strong>Guoen Hu</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 133--150.</p><p><strong>Abstract:</strong><br/>
Let $T_{1}$ , $T_{2}$ be two Calderón–Zygmund operators, and let $T_{1,b}$ be the commutator of $T_{1}$ with symbol $b\in\operatorname{BMO}(\mathbb{R}^{n})$ . In this article, we establish the quantitative weighted bounds on $L^{p}(\mathbb{R}^{n},w)$ with $w\in A_{p}(\mathbb{R}^{n})$ for the composite operator $T_{1,b}T_{2}$ .
</p>projecteuclid.org/euclid.bjma/1540454497_20181218040349Tue, 18 Dec 2018 04:03 ESTSpectral picture for rationally multicyclic subnormal operatorshttps://projecteuclid.org/euclid.bjma/1538121809<strong>Liming Yang</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 151--173.</p><p><strong>Abstract:</strong><br/>
For a pure bounded rationally cyclic subnormal operator $S$ on a separable complex Hilbert space $\mathcal{H}$ , Conway and Elias showed that $\operatorname{clos}(\sigma(S)\setminus\sigma_{e}(S))=\operatorname{clos}(\operatorname{Int}(\sigma(S)))$ . This article examines the property for rationally multicyclic ( $N$ -cyclic) subnormal operators. We show that there exists a $2$ -cyclic irreducible subnormal operator $S$ with $\operatorname{clos}(\sigma(S)\setminus\sigma_{e}(S))\neq\operatorname{clos}(\operatorname{Int}(\sigma(S)))$ . We also show the following. For a pure rationally $N$ -cyclic subnormal operator $S$ on $\mathcal{H}$ with the minimal normal extension $M$ on $\mathcal{K}\supset\mathcal{H}$ , let $\mathcal{K}_{m}=\operatorname{clos}(\operatorname{span}\{(M^{*})^{k}x:x\in\mathcal{H},0\le k\le m\}$ . Suppose that $M|_{\mathcal{K}_{N-1}}$ is pure. Then $\operatorname{clos}(\sigma(S)\setminus\sigma_{e}(S))=\operatorname{clos}(\operatorname{Int}(\sigma(S)))$ .
</p>projecteuclid.org/euclid.bjma/1538121809_20181218040349Tue, 18 Dec 2018 04:03 ESTOn some geometric properties of operator spaceshttps://projecteuclid.org/euclid.bjma/1543914019<strong>Arpita Mal</strong>, <strong>Debmalya Sain</strong>, <strong>Kallol Paul</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 174--191.</p><p><strong>Abstract:</strong><br/>
In this article, we study some geometric properties like parallelism, orthogonality, and semirotundity in the space of bounded linear operators. We completely characterize parallelism of two compact linear operators between normed linear spaces $\mathbb{X}$ and $\mathbb{Y}$ , assuming $\mathbb{X}$ to be reflexive. We also characterize parallelism of two bounded linear operators between normed linear spaces $\mathbb{X}$ and $\mathbb{Y}$ . We investigate parallelism and approximate parallelism in the space of bounded linear operators defined on a Hilbert space. Using the characterization of operator parallelism, we study Birkhoff–James orthogonality in the space of compact linear operators as well as bounded linear operators. Finally, we introduce the concept of semirotund points (semirotund spaces) which generalizes the notion of exposed points (strictly convex spaces). We further study semirotund operators and prove that $\mathbb{B}(\mathbb{X},\mathbb{Y})$ is a semirotund space which is not strictly convex if $\mathbb{X},\mathbb{Y}$ are finite-dimensional Banach spaces and $\mathbb{Y}$ is strictly convex.
</p>projecteuclid.org/euclid.bjma/1543914019_20181218040349Tue, 18 Dec 2018 04:03 ESTTranslation theorems for the Fourier–Feynman transform on the product function space $C_{a,b}^{2}[0,T]$https://projecteuclid.org/euclid.bjma/1544086815<strong>Seung Jun Chang</strong>, <strong>Jae Gil Choi</strong>, <strong>David Skoug</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 192--216.</p><p><strong>Abstract:</strong><br/>
In this article, we establish the Cameron–Martin translation theorems for the analytic Fourier–Feynman transform of functionals on the product function space $C_{a,b}^{2}[0,T]$ . The function space $C_{a,b}[0,T]$ is induced by the generalized Brownian motion process associated with continuous functions $a(t)$ and $b(t)$ on the time interval $[0,T]$ . The process used here is nonstationary in time and is subject to a drift $a(t)$ . To study our translation theorem, we introduce a Fresnel-type class $\mathcal{F}_{A_{1},A_{2}}^{a,b}$ of functionals on $C_{a,b}^{2}[0,T]$ , which is a generalization of the Kallianpur and Bromley–Fresnel class $\mathcal{F}_{A_{1},A_{2}}$ . We then proceed to establish the translation theorems for the functionals in $\mathcal{F}_{A_{1},A_{2}}^{a,b}$ .
</p>projecteuclid.org/euclid.bjma/1544086815_20181218040349Tue, 18 Dec 2018 04:03 ESTOn Hardy-type inequalities for weighted meanshttps://projecteuclid.org/euclid.bjma/1542358830<strong>Zsolt Páles</strong>, <strong>Paweł Pasteczka</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 217--233.</p><p><strong>Abstract:</strong><br/>
Our aim in this article is to establish weighted Hardy-type inequalities in a broad family of means. In other words, for a fixed vector of weights $(\lambda_{n})_{n=1}^{\infty}$ and a weighted mean $\mathscr{M}$ , we search for the smallest extended real number $C$ such that
\[\sum_{n=1}^{\infty}\lambda_{n}\mathscr{M}((x_{1},\ldots ,x_{n}),(\lambda_{1},\ldots,\lambda_{n}))\le C\sum_{n=1}^{\infty}\lambda_{n}x_{n}\quad \text{for all }x\in \ell_{1}(\lambda).\] The main results provide a complete answer in the case when $\mathscr{M}$ is monotone and satisfies the weighted counterpart of the Kedlaya inequality. In particular, this is the case if $\mathscr{M}$ is symmetric, concave, and the sequence $(\frac{\lambda_{n}}{\lambda_{1}+\cdots+\lambda_{n}})_{n=1}^{\infty}$ is nonincreasing. In addition, we prove that if $\mathscr{M}$ is a symmetric and monotone mean, then the biggest possible weighted Hardy constant is achieved if $\lambda$ is the constant vector.
</p>projecteuclid.org/euclid.bjma/1542358830_20181218040349Tue, 18 Dec 2018 04:03 ESTMultilinear operators factoring through Hilbert spaceshttps://projecteuclid.org/euclid.bjma/1542358829<strong>M. Fernández-Unzueta</strong>, <strong>S. García-Hernández</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 1, 234--254.</p><p><strong>Abstract:</strong><br/>
We characterize those bounded multilinear operators that factor through a Hilbert space in terms of its behavior in finite sequences. This extends a result, essentially due to Kwapień, from the linear to the multilinear setting. We prove that Hilbert–Schmidt and Lipschitz $2$ -summing multilinear operators naturally factor through a Hilbert space. We also prove that the class $\Gamma$ of all multilinear operators that factor through a Hilbert space is a maximal multi-ideal; moreover, we give an explicit formulation of a finitely generated tensor norm $\gamma$ which is in duality with $\Gamma$ .
</p>projecteuclid.org/euclid.bjma/1542358829_20181218040349Tue, 18 Dec 2018 04:03 ESTEnergy functional of the Volterra operatorhttps://projecteuclid.org/euclid.bjma/1548471622<strong>Yu-Xia Liang</strong>, <strong>Rongwei Yang</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 255--274.</p><p><strong>Abstract:</strong><br/>
We define the energy functional $E_{f,A}$ for a bounded linear operator $A$ acting on a Hilbert space ${\mathcal{H}}$ through a newly defined non-Euclidean metric $g_{f}(z)|dz|^{2}$ on its resolvent set $\rho (A)$ , where the vector $f\in \mathcal{H}$ . We investigate the extremal values of $E_{f,A}$ with respect to the change of $f$ . We conduct an in-depth study of the case when $A$ is the classical Volterra operator $V$ on $L^{2}([0,1])$ . Our main theorem suggests a likely connection between the energy functional and the invariant subspace problem.
</p>projecteuclid.org/euclid.bjma/1548471622_20190322040122Fri, 22 Mar 2019 04:01 EDTDual truncated Toeplitz $C^{*}$ -algebrashttps://projecteuclid.org/euclid.bjma/1548471621<strong>Yuanqi Sang</strong>, <strong>Yueshi Qin</strong>, <strong>Xuanhao Ding</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 275--292.</p><p><strong>Abstract:</strong><br/>
We establish the short exact sequences associated with the algebras generated by dual truncated Toeplitz operators on the orthogonal complement of the model space $K^{2}_{u}$ , and discuss spectral properties of dual truncated Toeplitz operators.
</p>projecteuclid.org/euclid.bjma/1548471621_20190322040122Fri, 22 Mar 2019 04:01 EDTOn the existence of solutions of variational inequalities in nonreflexive Banach spaceshttps://projecteuclid.org/euclid.bjma/1548666054<strong>Vy Khoi Le</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 293--313.</p><p><strong>Abstract:</strong><br/>
We are concerned in this article with an existence theorem for variational inequalities in nonreflexive Banach spaces with a general coercivity condition. The variational inequalities contain multivalued generalized pseudomonotone mappings and convex functionals, the nonreflexive Banach spaces form a complementary system, and the coercivity condition involves both the mapping and the functional. As an application, we study second-order elliptic variational inequalities with multivalued lower-order terms in general Orlicz–Sobolev spaces.
</p>projecteuclid.org/euclid.bjma/1548666054_20190322040122Fri, 22 Mar 2019 04:01 EDTToeplitz operators on Jordan–Kepler varietieshttps://projecteuclid.org/euclid.bjma/1548990185<strong>Harald Upmeier</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 314--337.</p><p><strong>Abstract:</strong><br/>
We study Toeplitz operators on Hilbert spaces of holomorphic functions on algebraic varieties, the generalized Kepler varieties defined in Jordan theoretic terms. Using the fine analysis of the reproducing kernel function, we construct and classify the irreducible representations of the $C^{*}$ -algebra generated by these operators (with smooth symbols). The limit behavior of Bergman-type measures under peaking functions is of special importance.
</p>projecteuclid.org/euclid.bjma/1548990185_20190322040122Fri, 22 Mar 2019 04:01 EDTA field-theoretic operator model and Cowen–Douglas classhttps://projecteuclid.org/euclid.bjma/1548666055<strong>Björn Gustafsson</strong>, <strong>Mihai Putinar</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 338--358.</p><p><strong>Abstract:</strong><br/>
In resonance with a recent geometric framework proposed by Douglas and Yang, we develop a functional model for certain linear bounded operators with rank-one self-commutator acting on a Hilbert space. By taking advantage of the refined existing theory of the principal function of a hyponormal operator, we transfer the whole action outside the spectrum, on the resolvent of the underlying operator, localized at a distinguished vector. The whole construction turns out to rely on an elementary algebra body involving analytic multipliers and Cauchy transforms. We propose a natural field theory interpretation of the resulting resolvent functional model.
</p>projecteuclid.org/euclid.bjma/1548666055_20190322040122Fri, 22 Mar 2019 04:01 EDTComplete systems of unitary invariants for some classes of $2$ -isometrieshttps://projecteuclid.org/euclid.bjma/1548990189<strong>Akash Anand</strong>, <strong>Sameer Chavan</strong>, <strong>Zenon Jan Jabłoński</strong>, <strong>Jan Stochel</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 359--385.</p><p><strong>Abstract:</strong><br/>
We characterize the unitary equivalence of $2$ -isometric operators satisfying the so-called kernel condition . This relies on a model for such operators built on operator-valued unilateral weighted shifts and on a characterization of the unitary equivalence of operator-valued unilateral weighted shifts in a fairly general context. We also provide a complete system of unitary invariants for $2$ -isometric weighted shifts on rooted directed trees satisfying the kernel condition. This is formulated purely in the language of graph theory—namely, in terms of certain generation branching degrees. Finally, we study the membership of the Cauchy dual operators of $2$ -isometries in classes $C_{0\cdot }$ and $C_{\cdot 0}$ .
</p>projecteuclid.org/euclid.bjma/1548990189_20190322040122Fri, 22 Mar 2019 04:01 EDTBounded Toeplitz operators on Bergman spacehttps://projecteuclid.org/euclid.bjma/1548061220<strong>Fugang Yan</strong>, <strong>Dechao Zheng</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 386--406.</p><p><strong>Abstract:</strong><br/>
In this paper, we obtain a sufficient condition for Toeplitz operators with locally integrable symbols to be bounded and compact on the Bergman space $L_{a}^{p}$ with $1\lt p\lt \infty$ in terms of generalized Carleson squares.
</p>projecteuclid.org/euclid.bjma/1548061220_20190322040122Fri, 22 Mar 2019 04:01 EDTFunctional characterizations of trace spaces in Lipschitz domainshttps://projecteuclid.org/euclid.bjma/1550048427<strong>Soumia Touhami</strong>, <strong>Abdellatif Chaira</strong>, <strong>Delfim F. M. Torres</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 407--426.</p><p><strong>Abstract:</strong><br/>
Using a factorization theorem of Douglas, we prove functional characterizations of trace spaces $H^{s}(\partial \Omega )$ involving a family of positive self-adjoint operators. Our method is based on the use of a suitable operator by taking the trace on the boundary $\partial \Omega $ of a bounded Lipschitz domain $\Omega \subset\mathbb{R}^{d}$ and applying Moore–Penrose pseudoinverse properties together with a special inner product on $H^{1}(\Omega )$ . We also establish generalized results of the Moore–Penrose pseudoinverse.
</p>projecteuclid.org/euclid.bjma/1550048427_20190322040122Fri, 22 Mar 2019 04:01 EDTInterpolation of $S$ -numbers and entropy numbers of operatorshttps://projecteuclid.org/euclid.bjma/1551258125<strong>Mieczysław Mastyło</strong>, <strong>Radosław Szwedek</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 427--448.</p><p><strong>Abstract:</strong><br/>
We introduce the notion of $\vec{s}$ -numbers for operators in Banach couples. We investigate variants of the approximation, Gelfand, and Kolmogorov numbers. In particular, we derive upper estimates of these numbers for operators between spaces generated by interpolation functors on Banach couples satisfying interpolation variants of approximation properties. We also study two-sided interpolation of entropy numbers.
</p>projecteuclid.org/euclid.bjma/1551258125_20190322040122Fri, 22 Mar 2019 04:01 EDTOn the Lax–Phillips scattering matrix of the abstract wave equationhttps://projecteuclid.org/euclid.bjma/1551150772<strong>M. Gawlik</strong>, <strong>A. Główczyk</strong>, <strong>S. Kuzhel</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 449--467.</p><p><strong>Abstract:</strong><br/>
We study the dependence of singularities of scattering matrices of the abstract wave equation on the choice of asymptotically equivalent outgoing/incoming subspaces. We apply the obtained results to the radial wave equation with nonlocal potential. In the latter case, the concept of associated inner function—introduced by Douglas, Shapiro, and Shields in 1970—plays an essential role.
</p>projecteuclid.org/euclid.bjma/1551150772_20190322040122Fri, 22 Mar 2019 04:01 EDTSolid cores and solid hulls of weighted Bergman spaceshttps://projecteuclid.org/euclid.bjma/1551150773<strong>José Bonet</strong>, <strong>Wolfgang Lusky</strong>, <strong>Jari Taskinen</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 468--485.</p><p><strong>Abstract:</strong><br/>
We determine the solid hull for $2\lt p\lt \infty$ and the solid core for $1\lt p\lt 2$ of weighted Bergman spaces $A_{\mu}^{p},1\lt p\lt \infty$ , of analytic functions on the disk and on the whole complex plane, for a very general class of nonatomic positive bounded Borel measures $\mu$ . New examples are presented. Moreover, we show that the space $A_{\mu}^{p},1\lt p\lt \infty$ , is solid if and only if the monomials are an unconditional basis of this space.
</p>projecteuclid.org/euclid.bjma/1551150773_20190322040122Fri, 22 Mar 2019 04:01 EDTWandering subspace property for homogeneous invariant subspaceshttps://projecteuclid.org/euclid.bjma/1552442837<strong>Jörg Eschmeier</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 2, 486--505.</p><p><strong>Abstract:</strong><br/>
For shift-like commuting tuples $T\in B(H)^{n}$ on graded Hilbert spaces $H$ , we show that each homogeneous invariant subspace $M$ of $T$ has finite index and is generated by its wandering subspace. Under suitable conditions on the grading $(H_{k})_{k\geq0}$ of $H$ , the algebraic direct sum $\tilde{M}=\bigoplus_{k\geq0}M\cap H_{k}$ becomes a finitely generated module over the polynomial ring $\mathbb{C}[z]$ in $n$ complex variables. We show that the wandering subspace $W_{T}(M)$ of $M$ is contained in $\tilde{M}$ and that each linear basis of $W_{T}(M)$ forms a minimal set of generators for $\tilde{M}$ . We describe an algorithm that transforms each set of homogeneous generators of $\tilde{M}$ into a minimal set of generators and can be used to compute minimal sets of generators for homogeneous ideals $I\subset\mathbb{C}[z]$ . We prove that each finitely generated $\gamma$ -graded commuting row contraction $T\in B(H)^{n}$ admits a finite weak resolution in the sense of Arveson or of Douglas and Misra.
</p>projecteuclid.org/euclid.bjma/1552442837_20190322040122Fri, 22 Mar 2019 04:01 EDTRank-one perturbations and Anderson-type Hamiltonianshttps://projecteuclid.org/euclid.bjma/1551258124<strong>Constanze Liaw</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 3, 507--523.</p><p><strong>Abstract:</strong><br/>
Motivated by applications of the discrete random Schrödinger operator, mathematical physicists and analysts began studying more general Anderson-type Hamiltonians; that is, the family of self-adjoint operators \begin{equation*}H_{\omega }=H+V_{\omega }\end{equation*} on a separable Hilbert space $\mathcal{H}$ , where the perturbation is given by \begin{equation*}V_{\omega }=\sum _{n}\omega _{n}(\cdot ,\varphi_{n})\varphi _{n}\end{equation*} with a sequence $\{\varphi _{n}\}\subset \mathcal{H}$ and independent identically distributed random variables $\omega _{n}$ . We show that the essential parts of Hamiltonians associated to any two realizations of the random variable are (almost surely) related by a rank-one perturbation. This result connects one of the least trackable perturbation problem (with almost surely noncompact perturbations) with one where the perturbation is “only” of rank-one perturbations. The latter presents a basic application of model theory. We also show that the intersection of the essential spectrum with open sets is almost surely either the empty set, or it has nonzero Lebesgue measure.
</p>projecteuclid.org/euclid.bjma/1551258124_20190709220348Tue, 09 Jul 2019 22:03 EDTAbelian theorems for distributional Kontorovich–Lebedev and Mehler–Fock transforms of general orderhttps://projecteuclid.org/euclid.bjma/1552464163<strong>Benito J. González</strong>, <strong>Emilio R. Negrín</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 3, 524--537.</p><p><strong>Abstract:</strong><br/>
Our goal in this article is to derive Abelian theorems for the Kontorovich–Lebedev and Mehler–Fock transforms of general order over distributions of compact support and over certain spaces of generalized functions.
</p>projecteuclid.org/euclid.bjma/1552464163_20190709220348Tue, 09 Jul 2019 22:03 EDTFactorized sectorial relations, their maximal-sectorial extensions, and form sumshttps://projecteuclid.org/euclid.bjma/1558749978<strong>Seppo Hassi</strong>, <strong>Adrian Sandovici</strong>, <strong>Henk de Snoo</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 3, 538--564.</p><p><strong>Abstract:</strong><br/>
In this paper we consider sectorial operators, or more generally, sectorial relations and their maximal-sectorial extensions in a Hilbert space ${\mathfrak{H}}$ . Our particular interest is in sectorial relations $S$ , which can be expressed in the factorized form \begin{equation*}S=T^{*}(I+iB)T\qquad \text{or}\qquad S=T(I+iB)T^{*},\end{equation*} where $B$ is a bounded self-adjoint operator in a Hilbert space ${\mathfrak{K}}$ and $T:{\mathfrak{H}}\to {\mathfrak{K}}$ (or $T:{\mathfrak{K}}\to {\mathfrak{H}}$ , respectively) is a linear operator or a linear relation which is not assumed to be closed. Using the specific factorized form of $S$ , a description of all the maximal-sectorial extensions of $S$ is given, along with a straightforward construction of the extreme extensions $S_{F}$ , the Friedrichs extension, and $S_{K}$ , the Kreĭn extension of $S$ , which uses the above factorized form of $S$ . As an application of this construction, we also treat the form sum of maximal-sectorial extensions of two sectorial relations.
</p>projecteuclid.org/euclid.bjma/1558749978_20190709220348Tue, 09 Jul 2019 22:03 EDTThe Bass and topological stable ranks for algebras of almost periodic functions on the real line, IIhttps://projecteuclid.org/euclid.bjma/1551409236<strong>Raymond Mortini</strong>, <strong>Amol Sasane</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 3, 565--581.</p><p><strong>Abstract:</strong><br/>
Let $\Lambda$ be either a subgroup of the integers ${\mathbb{Z}}$ , a semigroup in ${\mathbb{N}}$ , or $\Lambda={\mathbb{Q}}$ (resp., ${\mathbb{Q}}^{+}$ ). We determine the Bass and topological stable ranks of the algebras $\mathrm{AP}_{\Lambda}=\{f\in\mathrm{AP}:\sigma(f)\subseteq\Lambda\}$ of almost periodic functions on the real line and with Bohr spectrum in $\Lambda$ . This answers a question in the first part of this series of articles under the same heading, where it was shown that, in contrast to the present situation, these ranks were infinite for each semigroup $\Lambda$ of real numbers for which the ${\mathbb{Q}}$ -vector space generated by $\Lambda$ had infinite dimension.
</p>projecteuclid.org/euclid.bjma/1551409236_20190709220348Tue, 09 Jul 2019 22:03 EDTGeneralized quasidiagonality for extensionshttps://projecteuclid.org/euclid.bjma/1558749977<strong>P. W. Ng</strong>, <strong>Tracy Robin</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 3, 582--598.</p><p><strong>Abstract:</strong><br/>
We generalize the notion of quasidiagonality, for extensions, allowing for the case where the canonical ideal has few projections. We prove that the pointwise-norm limit of generalized quasidiagonal extensions is generalized quasidiagonal. We also provide a $K$ -theory sufficient condition for generalized quasidiagonality of certain extensions of simple continuous-scale $C^{*}$ -algebras, including certain continuous-scale hereditary $C^{*}$ -subalgebras of the stabilized Jiang–Su algebra.
</p>projecteuclid.org/euclid.bjma/1558749977_20190709220348Tue, 09 Jul 2019 22:03 EDTOn the property $\mathit{IR}$ of Friis and Rørdamhttps://projecteuclid.org/euclid.bjma/1559289617<strong>Lawrence G. Brown</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 3, 599--611.</p><p><strong>Abstract:</strong><br/>
Lin solved a longstanding problem as follows. For each $\epsilon \gt 0$ , there is $\delta \gt 0$ such that, if $h$ and $k$ are self-adjoint contractive $n\times n$ matrices and $\|hk-kh\|\lt \delta $ , then there are commuting self-adjoint matrices $h'$ and $k'$ such that $\|h'-h\|$ , $\|k'-k\|\lt \epsilon $ . Here $\delta $ depends only on $\epsilon $ and not on $n$ . Friis and Rørdam greatly simplified Lin’s proof by using a property they called $\mathit{IR}$ . They also generalized Lin’s result by showing that the matrix algebras can be replaced by any $C^{*}$ -algebras satisfying $\mathit{IR}$ . The purpose of this paper is to study the property $\mathit{IR}$ . One of our results shows how $\mathit{IR}$ behaves for $C^{*}$ -algebra extensions. Other results concern nonstable $K$ -theory. One shows that $\mathit{IR}$ (at least the stable version) implies a cancellation property for projections which is intermediate between the strong cancellation satisfied by $C^{*}$ -algebras of stable rank $1$ and the weak cancellation defined in a 2014 paper by Pedersen and the author.
</p>projecteuclid.org/euclid.bjma/1559289617_20190709220348Tue, 09 Jul 2019 22:03 EDTKernels of Hankel operators on the Hardy space over the bidiskhttps://projecteuclid.org/euclid.bjma/1560996142<strong>Kei Ji Izuchi</strong>, <strong>Kou Hei Izuchi</strong>, <strong>Yuko Izuchi</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 3, 612--626.</p><p><strong>Abstract:</strong><br/>
For a Hankel operator $H_{\xi }$ , $\xi \in L^{\infty }({\mathbb{T}}^{2})$ , on the Hardy space $H^{2}$ over the bidisk, $\operatorname{ker}H_{\overline{\xi }}$ is an invariant subspace of $H^{2}$ . It is known that there is an invariant subspace $M$ such that $\operatorname{ker}H_{\overline{\xi }}\neq M$ for every $\xi \in L^{\infty }$ . Let $\eta \in H^{\infty }$ be a nonconstant function. It is proved that if $\eta \perp \varphi (z)H^{2}+\psi (w)H^{2}$ for Blaschke products $\varphi (z)$ and $\psi (w)$ , then $\operatorname{ker}H_{\overline{\eta }}=\theta _{1}(z)\theta _{2}(w)H^{2}$ for some subproducts $\theta _{1}(z)$ and $\theta _{2}(w)$ of $\varphi (z)$ and $\psi (w)$ , respectively. If $\eta $ is $*$ -cyclic, then it is easy to see that $\operatorname{ker}H_{\overline{\eta }}=\{0\}$ . We give some examples $\eta $ satisfying $\operatorname{ker}H_{\overline{\eta }}=\{0\}$ but $\eta $ is not $*$ -cyclic.
</p>projecteuclid.org/euclid.bjma/1560996142_20190709220348Tue, 09 Jul 2019 22:03 EDTThe polar decomposition for adjointable operators on Hilbert $C^{*}$ -modules and $n$ -centered operatorshttps://projecteuclid.org/euclid.bjma/1558749979<strong>Na Liu</strong>, <strong>Wei Luo</strong>, <strong>Qingxiang Xu</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 3, 627--646.</p><p><strong>Abstract:</strong><br/>
Let $n$ be any natural number. The $n$ -centered operator is introduced for adjointable operators on Hilbert $C^{*}$ -modules. Based on the characterizations of the polar decomposition for the product of two adjointable operators, $n$ -centered operators, centered operators as well as binormal operators are clarified, and some results known for the Hilbert space operators are improved. It is proved that for an adjointable operator $T$ , if $T$ is Moore–Penrose invertible and is $n$ -centered, then its Moore–Penrose inverse is also $n$ -centered. A Hilbert space operator $T$ is constructed such that $T$ is $n$ -centered, whereas it fails to be $(n+1)$ -centered.
</p>projecteuclid.org/euclid.bjma/1558749979_20190709220348Tue, 09 Jul 2019 22:03 EDTOn the structure of universal functions for classes $L^{p}[0,1)^{2},p\in (0,1)$ , with respect to the double Walsh systemhttps://projecteuclid.org/euclid.bjma/1560996143<strong>Martin Grigoryan</strong>, <strong>Artsrun Sargsyan</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 3, 647--674.</p><p><strong>Abstract:</strong><br/>
We address questions on the existence and structure of universal functions for classes $L^{p}[0,1)^{2}$ , $p\in (0,1)$ , with respect to the double Walsh system. It is shown that there exists a measurable set $E\subset [0,1)^{2}$ with measure arbitrarily close to $1$ , such that, by a proper modification of any integrable function $f\in L^{1}[0,1)^{2}$ outside $E$ , we can get an integrable function $\tilde{f}\in L^{1}[0,1)^{2}$ , which is universal for each class $L^{p}[0,1)^{2}$ , $p\in (0,1)$ , with respect to the double Walsh system in the sense of signs of Fourier coefficients.
</p>projecteuclid.org/euclid.bjma/1560996143_20190709220348Tue, 09 Jul 2019 22:03 EDTBoundedness of Cesàro and Riesz means in variable dyadic Hardy spaceshttps://projecteuclid.org/euclid.bjma/1559268029<strong>Kristóf Szarvas</strong>, <strong>Ferenc Weisz</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 3, 675--696.</p><p><strong>Abstract:</strong><br/>
We consider two types of maximal operators. We prove that, under some conditions, each maximal operator is bounded from the classical dyadic martingale Hardy space $H_{p}$ to the classical Lebesgue space $L_{p}$ and from the variable dyadic martingale Hardy space $H_{p(\cdot )}$ to the variable Lebesgue space $L_{p(\cdot )}$ . Using this, we can prove the boundedness of the Cesàro and Riesz maximal operator from $H_{p(\cdot )}$ to $L_{p(\cdot )}$ and from the variable Hardy–Lorentz space $H_{p(\cdot ),q}$ to the variable Lorentz space $L_{p(\cdot ),q}$ . As a consequence, we can prove theorems about almost everywhere and norm convergence.
</p>projecteuclid.org/euclid.bjma/1559268029_20190709220348Tue, 09 Jul 2019 22:03 EDTRiesz transforms, Cauchy–Riemann systems, and Hardy-amalgam spaceshttps://projecteuclid.org/euclid.bjma/1558749976<strong>Al-Tarazi Assaubay</strong>, <strong>Jorge J. Betancor</strong>, <strong>Alejandro J. Castro</strong>, <strong>Juan C. Fariña</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 3, 697--725.</p><p><strong>Abstract:</strong><br/>
In this article we study Hardy spaces $\mathcal{H}^{p,q}(\mathbb{R}^{d})$ , $0\lt p,q\lt \infty $ , modeled over amalgam spaces $(L^{p},\ell ^{q})(\mathbb{R}^{d})$ . We characterize $\mathcal{H}^{p,q}(\mathbb{R}^{d})$ by using first-order classical Riesz transforms and compositions of first-order Riesz transforms, depending on the values of the exponents $p$ and $q$ . Also, we describe the distributions in $\mathcal{H}^{p,q}(\mathbb{R}^{d})$ as the boundary values of solutions of harmonic and caloric Cauchy–Riemann systems. We remark that caloric Cauchy–Riemann systems involve fractional derivatives in the time variable. Finally, we characterize the functions in $L^{2}(\mathbb{R}^{d})\cap \mathcal{H}^{p,q}(\mathbb{R}^{d})$ by means of Fourier multipliers $m_{\theta }$ with symbol $\theta (\cdot /|\cdot |)$ , where $\theta \in C^{\infty }(\mathbb{S}^{d-1})$ and $\mathbb{S}^{d-1}$ denotes the unit sphere in $\mathbb{R}^{d}$ .
</p>projecteuclid.org/euclid.bjma/1558749976_20190709220348Tue, 09 Jul 2019 22:03 EDTPositivity of $2\times 2$ block matrices of operatorshttps://projecteuclid.org/euclid.bjma/1558145239<strong>Mohammad Sal Moslehian</strong>, <strong>Mohsen Kian</strong>, <strong>Qingxiang Xu</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 3, 726--743.</p><p><strong>Abstract:</strong><br/>
We review some significant generalizations and applications of the celebrated Douglas theorem on equivalence of factorization, range inclusion, and majorization of operators. We then apply it to find a characterization of the positivity of $2\times 2$ block matrices of operators on Hilbert spaces. Finally, we describe the nature of such block matrices and provide several ways for showing their positivity.
</p>projecteuclid.org/euclid.bjma/1558145239_20190709220348Tue, 09 Jul 2019 22:03 EDTSolutions to fractional Sobolev-type integro-differential equations in Banach spaces with operator pairs and impulsive conditionshttps://projecteuclid.org/euclid.bjma/1570608161<strong>Jin Liang</strong>, <strong>Yunyi Mu</strong>, <strong>Ti-Jun Xiao</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 4, 745--768.</p><p><strong>Abstract:</strong><br/>
We are concerned with fractional Sobolev-type integro-differential equations in Banach spaces with operator pairs and impulsive conditions, where the operator pairs generate propagation families. With the help of the theory of propagation family and Laplace transforms, along with an estimate for a special sequence improved in this article, we introduce a definition of mild solutions to the impulsive problem for these abstract fractional Sobolev-type integro-differential equations and we establish general existence theorems and a continuous dependence theorem, which essentially extend some previous conclusions. In our results, the operator $B$ could be unbounded, and the existence of an operator $B^{-1}$ is not necessarily needed. Moreover, we give some examples to illustrate our main results.
</p>projecteuclid.org/euclid.bjma/1570608161_20191009040301Wed, 09 Oct 2019 04:03 EDTVariable Hardy–Lorentz spaces associated to operators satisfying Davies–Gaffney estimateshttps://projecteuclid.org/euclid.bjma/1570608162<strong>Yahui Zuo</strong>, <strong>Khedoudj Saibi</strong>, <strong>Yong Jiao</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 4, 769--797.</p><p><strong>Abstract:</strong><br/>
Let $L$ be a one-to-one operator of type $w$ with $w\in [0,\pi /2)$ , which satisfies the Davies–Gaffney estimates and has a bounded holomorphic calculus, and let $p(\cdot )$ be a measurable function on ${\mathbb{R}}^{n}$ with $0\lt p_{-}:=\operatorname{ess}\inf _{x\in {{\mathbb{R}}}^{n}}p(x)\leq \operatorname{ess}\sup _{x\in {{\mathbb{R}}}^{n}}p(x)=:p_{+}\lt \infty $ . Under the assumption that $p(\cdot )$ satisfies the global log-Hölder condition, we introduce the variable Hardy–Lorentz space $H_{L}^{p(\cdot ),q}(\mathbb{R}^{n})$ for $0\lt q\lt \infty $ and construct its molecular decomposition. Furthermore, we investigate the dual spaces of the variable Hardy–Lorentz space $H_{L}^{p(\cdot ),q}(\mathbb{R}^{n})$ with $0\lt p_{-}\leq p_{+}\leq 1$ and $0\lt q\lt \infty $ . These results are new even when $p(\cdot )$ is a constant.
</p>projecteuclid.org/euclid.bjma/1570608162_20191009040301Wed, 09 Oct 2019 04:03 EDTInterpolation of Haagerup noncommutative Hardy spaceshttps://projecteuclid.org/euclid.bjma/1570608163<strong>Turdebek N. Bekjan</strong>, <strong>Madi Raikhan</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 4, 798--814.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{M}$ be a $\sigma $ -finite von Neumann algebra equipped with a normal faithful state $\varphi $ , and let $\mathcal{A}$ be a maximal subdiagonal algebra of $\mathcal{M}$ . We prove a Stein–Weiss-type interpolation theorem of Haagerup noncommutative $H^{p}$ -spaces associated with ${\mathcal{A}}$ .
</p>projecteuclid.org/euclid.bjma/1570608163_20191009040301Wed, 09 Oct 2019 04:03 EDTDisjoint hypercyclic weighted pseudoshift operators generated by different shiftshttps://projecteuclid.org/euclid.bjma/1570608164<strong>Ya Wang</strong>, <strong>Cui Chen</strong>, <strong>Ze-Hua Zhou</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 4, 815--836.</p><p><strong>Abstract:</strong><br/>
Let $I$ be a countably infinite index set, and let $X$ be a Banach sequence space over $I$ . In this article, we characterize the disjoint hypercyclic and supercyclic weighted pseudoshift operators on $X$ in terms of the weights, the OP-basis, and the shift mappings on $I$ . Also, the shifts on weighted $L^{p}$ spaces of a directed tree and the operator weighted shifts on $\ell ^{2}(\mathbb{Z},\mathcal{K})$ are investigated as special cases.
</p>projecteuclid.org/euclid.bjma/1570608164_20191009040301Wed, 09 Oct 2019 04:03 EDTExtensions of uniform algebrashttps://projecteuclid.org/euclid.bjma/1570608165<strong>Sam Morley</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 4, 837--863.</p><p><strong>Abstract:</strong><br/>
The theory of algebraic extensions of Banach algebras is well established, and there are many constructions which yield interesting extensions. In particular, Cole’s method for extending uniform algebras by adding square roots of functions to a given uniform algebra has been used to solve many problems within uniform algebra theory. However, there are numerous other examples in the theory of uniform algebras that can be realized as extensions of a uniform algebra, and these more general extensions have received little attention in the literature. In this paper, we investigate more general classes of uniform algebra extensions. We introduce a new class of extensions of uniform algebras, and show that several important properties of the original uniform algebra are preserved in these extensions. We also show that several well-known examples from the theory of uniform algebras belong to these more general classes of uniform algebra extensions.
</p>projecteuclid.org/euclid.bjma/1570608165_20191009040301Wed, 09 Oct 2019 04:03 EDTCarleson measures on circular domains and canonical embeddings of Hardy spaces into function latticeshttps://projecteuclid.org/euclid.bjma/1570608166<strong>Paweł Mleczko</strong>, <strong>Michał Rzeczkowski</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 4, 864--883.</p><p><strong>Abstract:</strong><br/>
We study general variants of spaces of holomorphic functions on circular domains on the complex plane. We define Hardy-type spaces generated by Banach function lattices, for which we prove the Carleson theorem. We also analyze canonical embeddings of such spaces into appropriate function lattices. Finally, we study composition operators on Hardy-type spaces on circular domains and characterize order-boundedness of such maps.
</p>projecteuclid.org/euclid.bjma/1570608166_20191009040301Wed, 09 Oct 2019 04:03 EDTAtomic characterizations of weak martingale Musielak–Orlicz Hardy spaces and their applicationshttps://projecteuclid.org/euclid.bjma/1570608167<strong>Guangheng Xie</strong>, <strong>Dachun Yang</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 4, 884--917.</p><p><strong>Abstract:</strong><br/>
Let $(\Omega ,\mathcal{F},\mathbb{P})$ be a probability space, and let $\varphi :\Omega \times [0,\infty )\to [0,\infty )$ be a Musielak–Orlicz function. In this article, we establish the atomic characterizations of weak martingale Musielak–Orlicz Hardy spaces $\mathit{WH}_{\varphi }^{s}(\Omega )$ , $\mathit{WH}_{\varphi }^{M}(\Omega )$ , $\mathit{WH}_{\varphi }^{S}(\Omega )$ , $WP_{\varphi }(\Omega )$ , and $WQ_{\varphi }(\Omega )$ . We then use these atomic characterizations to obtain the boundedness of $\sigma $ -sublinear operators from weak martingale Musielak–Orlicz Hardy spaces to weak Musielak–Orlicz spaces, as well as some martingale inequalities which further clarify the relationships among these weak martingale Musielak–Orlicz Hardy spaces. All these results improve and generalize the corresponding results on weak martingale Orlicz–Hardy spaces. Moreover, we improve all the known results on weak martingale Musielak–Orlicz Hardy spaces. In particular, both the boundedness of $\sigma $ -sublinear operators and the martingale inequalities, for weak weighted martingale Hardy spaces as well as for weak weighted martingale Orlicz–Hardy spaces, are new.
</p>projecteuclid.org/euclid.bjma/1570608167_20191009040301Wed, 09 Oct 2019 04:03 EDTOn uniform connectivity of algebraic matrix setshttps://projecteuclid.org/euclid.bjma/1570608168<strong>Fredy Vides</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 4, 918--943.</p><p><strong>Abstract:</strong><br/>
We study the uniform path connectivity of sets of matrix tuples that satisfy some additional constraints, and more specifically, given $\varepsilon \gt 0$ , a fixed metric $\eth $ in ${M_{n}(\mathbb{C})}^{m}$ induced by the operator norm $\|\cdot \|$ , any collection of $r$ nonconstant polynomials $p_{1}(x_{1},\ldots ,x_{m}),\ldots ,p_{r}(x_{1},\ldots ,x_{m})$ over $\mathbb{C}$ with finite zero set $\mathbf{Z}(p_{1},\ldots ,p_{r})\subset \mathbb{C}^{m}$ and any $m$ -tuple $\mathbf{X}=(X_{1},\ldots ,X_{m})$ in the set $\mathbb{ZD}_{n}^{m}(p_{1},\ldots ,p_{r})\subseteq M_{n}^{m}(\mathbb{C})$ of commuting normal matrix contractions such that $\|p_{j}(Y_{1},\ldots ,Y_{m})\|=0$ for each $(Y_{1},\ldots ,Y_{m})\in \mathbb{ZD}_{n}^{m}(p_{1},\ldots ,p_{r})$ and each $1\leq j\leq r$ . The author proves the existence of paths between arbitrary $m$ -tuples that belong to the intersection of $\mathbb{ZD}_{n}^{m}(p_{1},\ldots,p_{r})$ and the open $\delta $ -ball $B_{\eth }(\mathbf{X},\delta )$ centered at $\mathbf{X}$ for some $\delta \gt 0$ that can be chosen independently of $n$ . In addition, the author proves that the aforementioned paths are contained in the intersection of $B_{\eth }(\mathbf{X},\varepsilon )$ and $\mathbb{ZD}_{n}^{m}(p_{1},\ldots ,p_{r})$ . Some connections of the main results with structure-preserving perturbation theory and preconditioning techniques are outlined.
</p>projecteuclid.org/euclid.bjma/1570608168_20191009040301Wed, 09 Oct 2019 04:03 EDTOn solutions of an infinite system of nonlinear integral equations on the real half-axishttps://projecteuclid.org/euclid.bjma/1570608169<strong>Józef Banaś</strong>, <strong>Agnieszka Chlebowicz</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 4, 944--968.</p><p><strong>Abstract:</strong><br/>
We investigate the existence of solutions of an infinite system of integral equations of Volterra–Hammerstein type on the real half-axis. The method applied in our study is connected with the construction of a suitable measure of noncompactness in the space of continuous and bounded functions defined on the real half-axis with values in the space $c_{0}$ consisting of real sequences converging to zero and equipped with the classical supremum norm. We use the mentioned measure of noncompactness together with a fixed point theorem of Darbo type. Our investigations are illustrated by an example.
</p>projecteuclid.org/euclid.bjma/1570608169_20191009040301Wed, 09 Oct 2019 04:03 EDTIntrinsic square function characterizations of weak Musielak–Orlicz Hardy spaceshttps://projecteuclid.org/euclid.bjma/1570608170<strong>Xianjie Yan</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 4, 969--988.</p><p><strong>Abstract:</strong><br/>
Let $\varphi :\mathbb{R}^{n}\times [0,\infty )\to [0,\infty )$ satisfy that, for any given $x\in \mathbb{R}^{n}$ , $\varphi (x,\cdot )$ is an Orlicz function and $\varphi (\cdot ,t)$ is a Muckenhoupt $A_{\infty}(\mathbb{R}^{n})$ weight uniformly in $t\in (0,\infty )$ . In this article, via using the atomic and Littlewood–Paley function characterizations of the weak Musielak–Orlicz Hardy space $\mathit{WH}^{\varphi }(\mathbb{R}^{n})$ , for any $\alpha \in (0,1]$ and $s\in\mathbb{Z}_{+}$ , we first establish its $s$ -order intrinsic square function characterizations in terms of the intrinsic Lusin area function $S_{\alpha ,s}$ , the intrinsic $g$ -function $g_{\alpha ,s}$ and the intrinsic $g_{\lambda }^{*}$ -function $g_{\lambda ,\alpha ,s}^{*}$ with the best known range $\lambda \in (2+2(\alpha +s)/n,\infty )$ .
</p>projecteuclid.org/euclid.bjma/1570608170_20191009040301Wed, 09 Oct 2019 04:03 EDTSemi-Fredholm theory on Hilbert $C^{*}$ -moduleshttps://projecteuclid.org/euclid.bjma/1570608171<strong>Stefan Ivković</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 4, 989--1016.</p><p><strong>Abstract:</strong><br/>
We establish the semi-Fredholm theory on Hilbert $C^{*}$ -modules as a continuation of Fredholm theory on Hilbert $C^{*}$ -modules established by Mishchenko and Fomenko. We give a definition of a semi-Fredholm operator on Hilbert $C^{*}$ -module, and we prove that these semi-Fredholm operators are those that are one-sided invertible modulo compact operators, that the set of proper semi-Fredholm operators is open, and many other results that generalize their classical counterparts.
</p>projecteuclid.org/euclid.bjma/1570608171_20191009040301Wed, 09 Oct 2019 04:03 EDTWeak frames in Hilbert $C^{*}$ -modules with application in Gabor analysishttps://projecteuclid.org/euclid.bjma/1570608172<strong>Damir Bakić</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 13, Number 4, 1017--1075.</p><p><strong>Abstract:</strong><br/>
In the first part, we describe the dual $\ell ^{2}(\mathsf{A})^{\prime}$ of the standard Hilbert $C^{*}$ -module $\ell ^{2}(\mathsf{A})$ over an arbitrary (not necessarily unital) $C^{*}$ -algebra $\mathsf{A}$ . When $\mathsf{A}$ is a von Neumann algebra, this enables us to construct explicitly a self-dual Hilbert $\mathsf{A}$ -module $\ell ^{2}_{\mathrm{strong}}(\mathsf{A})$ that is isometrically isomorphic to $\ell ^{2}(\mathsf{A})^{\prime }$ , which contains $\ell ^{2}(\mathsf{A})$ , and whose $\mathsf{A}$ -valued inner product extends the original inner product on $\ell ^{2}(\mathsf{A})$ . This serves as a concrete realization of a general construction for Hilbert $C^{*}$ -modules over von Neumann algebras introduced by Paschke. Then we introduce a concept of a weak Bessel sequence and a weak frame in Hilbert $C^{*}$ -modules over von Neumann algebras. The dual $\ell ^{2}(\mathsf{A})^{\prime }$ is recognized as a suitable target space for the analysis operator. We describe fundamental properties of weak frames such as the correspondence with surjective adjointable operators, the canonical dual, the reconstruction formula, and so on, first for self-dual modules and then, working in the dual, for general modules. The last part describes a class of Hilbert $C^{*}$ -modules over $L^{\infty }(I)$ , where $I$ is a bounded interval on the real line, that appears naturally in connection with Gabor (i.e., Weyl–Heisenberg) systems. We then demonstrate that Gabor Bessel systems and Gabor frames in $L^{2}(\mathbb{R})$ are in a bijective correspondence with weak Bessel systems and weak frames of translates by $a$ in these modules over $L^{\infty }[0,\frac{1}{b}]$ , where $a,b\gt 0$ are the lattice parameters. In this setting new proofs of several classical results on Gabor frames are demonstrated and some new ones are obtained.
</p>projecteuclid.org/euclid.bjma/1570608172_20191009040301Wed, 09 Oct 2019 04:03 EDT