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 EDTLimit dynamical systems and $C^{*}$ -algebras from self-similar graph actionshttps://projecteuclid.org/euclid.bjma/1498118444<strong>Inhyeop Yi</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 4, 764--784.</p><p><strong>Abstract:</strong><br/>
In this article, we study dynamical and $C^{*}$ -algebraic properties of self-similar group actions on finite directed graphs. We investigate the structure of limit dynamical systems induced from group actions on graphs, and we deduce conditions of group actions and graphs for the groupoid $C^{*}$ -algebras defined by limit dynamical systems to be simple, separable, purely infinite, nuclear, and satisfying the universal coefficient theorem.
</p>projecteuclid.org/euclid.bjma/1498118444_20171004220208Wed, 04 Oct 2017 22:02 EDTAnalytic Fourier–Feynman transforms and convolution products associated with Gaussian processes on Wiener spacehttps://projecteuclid.org/euclid.bjma/1498809807<strong>Seung Jun Chang</strong>, <strong>Jae Gil Choi</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 4, 785--807.</p><p><strong>Abstract:</strong><br/>
Using Gaussian processes, we define a very general convolution product of functionals on Wiener space and we investigate fundamental relationships between the generalized Fourier–Feynman transforms and the generalized convolution products. Using two rotation theorems of Gaussian processes, we establish that both of the generalized Fourier–Feynman transform of the generalized convolution product and the generalized convolution product of the generalized Fourier–Feynman transforms of functionals on Wiener space are represented as products of the generalized Fourier–Feynman transforms of each functional, with examples.
</p>projecteuclid.org/euclid.bjma/1498809807_20171004220208Wed, 04 Oct 2017 22:02 EDTSine and cosine equations on hypergroupshttps://projecteuclid.org/euclid.bjma/1498809806<strong>Żywilla Fechner</strong>, <strong>László Székelyhidi</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 4, 808--824.</p><p><strong>Abstract:</strong><br/>
This article deals with trigonometric functional equations on hypergroups. We describe the general continuous solution of sine and cosine addition formulas and a so-called sine-cosine functional equation on a locally compact hypergroup in terms of exponential functions, sine functions, and second-order generalized moment functions.
</p>projecteuclid.org/euclid.bjma/1498809806_20171004220208Wed, 04 Oct 2017 22:02 EDTOn the composition ideals of Lipschitz mappingshttps://projecteuclid.org/euclid.bjma/1502935412<strong>Khalil Saadi</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 4, 825--840.</p><p><strong>Abstract:</strong><br/>
We study some properties of Lipschitz mappings which admit factorization through an operator ideal. Lipschitz cross norms have been established from known tensor norms in order to represent certain classes of Lipschitz mappings. Inspired by the definition of $p$ -summing linear operators, we derive a new class of Lipschitz mappings that is called strictly Lipschitz $p$ -summing .
</p>projecteuclid.org/euclid.bjma/1502935412_20171004220208Wed, 04 Oct 2017 22:02 EDTTent spaces at endpointshttps://projecteuclid.org/euclid.bjma/1502935413<strong>Yong Ding</strong>, <strong>Ting Mei</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 4, 841--863.</p><p><strong>Abstract:</strong><br/>
In 1985, Coifman, Meyer, and Stein gave the duality of the tent spaces; that is, $(T_{q}^{p}(\mathbb{R}^{n+1}_{+}))^{\ast}=T_{q'}^{p'}(\mathbb{R}^{n+1}_{+})$ for $1\lt p,q\lt \infty$ , and $(T_{\infty}^{1}(\mathbb{R}^{n+1}_{+}))^{\ast}=\mathscr{C}(\mathbb{R}^{n+1}_{+})$ , $(T_{q}^{1}(\mathbb{R}^{n+1}_{+}))^{\ast}=T_{q'}^{\infty}(\mathbb{R}^{n+1}_{+})$ for $1\lt q\lt \infty$ , where $\mathscr{C}(\mathbb{R}^{n+1}_{+})$ denotes the Carleson measure space on $\mathbb{R}^{n+1}_{+}$ . We prove that $(\mathscr{C}_{v}(\mathbb{R}^{n+1}_{+}))^{\ast}=T_{\infty}^{1}(\mathbb{R}^{n+1}_{+})$ , which we introduced recently, where $\mathscr{C}_{v}(\mathbb{R}^{n+1}_{+})$ is the vanishing Carleson measure space on $\mathbb{R}^{n+1}_{+}$ . We also give the characterizations of $T_{q}^{\infty}(\mathbb{R}^{n+1}_{+})$ by the boundedness of the Poisson integral. As application, we give the boundedness and compactness on $L^{q}(\mathbb{R}^{n})$ of the paraproduct $\pi_{F}$ associated with the tent space $T_{q}^{\infty}(\mathbb{R}^{n+1}_{+})$ , and we extend partially an interesting result given by Coifman, Meyer, and Stein, which establishes a close connection between the tent spaces $T_{2}^{p}(\mathbb{R}^{n+1}_{+})$ $(1\le p\le\infty)$ and $L^{p}(\mathbb{R}^{n})$ , $H^{p}(\mathbb{R}^{n})$ and $\mathit{BMO}(\mathbb{R}^{n})$ spaces.
</p>projecteuclid.org/euclid.bjma/1502935413_20171004220208Wed, 04 Oct 2017 22:02 EDTPoint multipliers and the Gleason–Kahane–Żelazko theoremhttps://projecteuclid.org/euclid.bjma/1502935411<strong>Razieh Sadat Ghodrat</strong>, <strong>Fereshteh Sady</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 4, 864--879.</p><p><strong>Abstract:</strong><br/>
Let $A$ be a Banach algebra, and let $\mathcal{X}$ be a left Banach $A$ -module. In this paper, using the notation of point multipliers on left Banach modules, we introduce a certain type of spectrum for the elements of $\mathcal{X}$ and we also introduce a certain subset of $\mathcal{X}$ which behaves as the set of invertible elements of a commutative unital Banach algebra. Among other things, we use these sets to give some Gleason–Kahane–Żelazko theorems for left Banach $A$ -modules.
</p>projecteuclid.org/euclid.bjma/1502935411_20171004220208Wed, 04 Oct 2017 22:02 EDTDuality properties for generalized frameshttps://projecteuclid.org/euclid.bjma/1503993619<strong>F. Enayati</strong>, <strong>M. S. Asgari</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 4, 880--898.</p><p><strong>Abstract:</strong><br/>
We introduce the concept of Riesz-dual sequences for g-frames. In this paper we show that, for any sequence of operators, we can construct a corresponding sequence of operators with a kind of duality relation between them. This construction is used to prove duality principles in g-frame theory, which can be regarded as general versions of several well-known duality principles for frames. We also derive a simple characterization of a g-Riesz basic sequence as a g-R-dual sequence of a g-frame in the tight case.
</p>projecteuclid.org/euclid.bjma/1503993619_20171004220208Wed, 04 Oct 2017 22:02 EDTCharacterizations of asymmetric truncated Toeplitz operatorshttps://projecteuclid.org/euclid.bjma/1504080092<strong>Crisina Câmara</strong>, <strong>Joanna Jurasik</strong>, <strong>Kamila Kliś-Garlicka</strong>, <strong>Marek Ptak</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 4, 899--922.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to investigate asymmetric truncated Toeplitz operators with $L^{2}$ -symbols between two different model spaces given by inner functions such that one divides the other. The class of symbols corresponding to the zero operator is described. Asymmetric truncated Toeplitz operators are characterized in terms of operators of rank at most $2$ , and the relations with the corresponding symbols are studied.
</p>projecteuclid.org/euclid.bjma/1504080092_20171004220208Wed, 04 Oct 2017 22:02 EDTNon-self-adjoint Schrödinger operators with nonlocal one-point interactionshttps://projecteuclid.org/euclid.bjma/1505116817<strong>Sergii Kuzhel</strong>, <strong>Miloslav Znojil</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 4, 923--944.</p><p><strong>Abstract:</strong><br/>
We generalize and study, within the framework of quantum mechanics and working with $1$ -dimensional, manifestly non-Hermitian Hamiltonians $H=-{d^{2}}/{dx^{2}}+V$ , the traditional class of exactly solvable models with local point interactions $V=V(x)$ . We discuss the consequences of the use of nonlocal point interactions such that $(Vf)(x)=\int K(x,s)f(s)\,ds$ by means of the suitably adapted formalism of boundary triplets.
</p>projecteuclid.org/euclid.bjma/1505116817_20171004220208Wed, 04 Oct 2017 22:02 EDTLinear dependency of translations and square-integrable representationshttps://projecteuclid.org/euclid.bjma/1504836178<strong>Peter A. Linnell</strong>, <strong>Michael J. Puls</strong>, <strong>Ahmed Roman</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 4, 945--962.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a locally compact group. We examine the problem of determining when nonzero functions in $L^{2}(G)$ have linearly independent left translations. In particular, we establish some results for the case when $G$ has an irreducible, square-integrable, unitary representation. We apply these results to the special cases of the affine group, the shearlet group, and the Weyl–Heisenberg group. We also investigate the case when $G$ has an abelian, closed subgroup of finite index.
</p>projecteuclid.org/euclid.bjma/1504836178_20171004220208Wed, 04 Oct 2017 22:02 EDTMultiplicative operator functions and abstract Cauchy problemshttps://projecteuclid.org/euclid.bjma/1515402092<strong>Felix Früchtl</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 27 pages.</p><p><strong>Abstract:</strong><br/>
We use the duality between functional and differential equations to solve several classes of abstract Cauchy problems related to special functions. As a general framework, we investigate operator functions which are multiplicative with respect to convolution of a hypergroup. This setting contains all representations of (hyper)groups, and properties of continuity are shown; examples are provided by translation operator functions on homogeneous Banach spaces and weakly stationary processes indexed by hypergroups. Then we show that the concept of a multiplicative operator function can be used to solve a variety of abstract Cauchy problems, containing discrete, compact, and noncompact problems, including $C_{0}$ -groups and cosine operator functions, and more generally, Sturm–Liouville operator functions.
</p>projecteuclid.org/euclid.bjma/1515402092_20180108040158Mon, 08 Jan 2018 04:01 ESTLower and upper local uniform $K$ -monotonicity in symmetric spaceshttps://projecteuclid.org/euclid.bjma/1513674116<strong>Maciej Ciesielski</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 17 pages.</p><p><strong>Abstract:</strong><br/>
Using the local approach to the global structure of a symmetric space $E$ , we establish a relationship between strict $K$ -monotonicity, lower (resp., upper) local uniform $K$ -monotonicity, order continuity, and the Kadec–Klee property for global convergence in measure. We also answer the question: Under which condition does upper local uniform $K$ -monotonicity coincide with upper local uniform monotonicity? Finally, we present a correlation between $K$ -order continuity and lower local uniform $K$ -monotonicity in a symmetric space $E$ under some additional assumptions on $E$ .
</p>projecteuclid.org/euclid.bjma/1513674116_20180108040158Mon, 08 Jan 2018 04:01 ESTRelatively compact sets in variable-exponent Lebesgue spaceshttps://projecteuclid.org/euclid.bjma/1513674117<strong>Rovshan Bandaliyev</strong>, <strong>Przemysław Górka</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 16 pages.</p><p><strong>Abstract:</strong><br/>
We study totally bounded sets in variable Lebesgue spaces. The full characterization of this kind of sets is given for the case of variable Lebesgue space on metric measure spaces. Furthermore, the sufficient conditions for compactness are shown without assuming $\log$ -Hölder continuity of the exponent.
</p>projecteuclid.org/euclid.bjma/1513674117_20180108040158Mon, 08 Jan 2018 04:01 ESTReducing subspaces for a class of nonanalytic Toeplitz operatorshttps://projecteuclid.org/euclid.bjma/1513674118<strong>Jia Deng</strong>, <strong>Yufeng Lu</strong>, <strong>Yanyue Shi</strong>, <strong>Yinyin Hu</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 25 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we give a uniform characterization for the reducing subspaces for $T_{\varphi}$ with the symbol $\varphi(z)=z^{k}+\bar{z}^{l}$ ( $k,l\in\mathbb{Z}_{+}^{2}$ ) on the Bergman spaces over the bidisk, including the known cases that $\varphi(z_{1},z_{2})=z_{1}^{N}z_{2}^{M}$ and $\varphi(z_{1},z_{2})=z_{1}^{N}+\overline{z}_{2}^{M}$ with $N,M\in\mathbb{Z}_{+}$ . Meanwhile, the reducing subspaces for $T_{z^{N}+\overline{z}^{M}}$ ( $N,M\in \mathbb{Z}_{+}$ ) on the Bergman space over the unit disk are also described. Finally, we state these results in terms of the commutant algebra $\mathcal{V}^{*}(\varphi)$ .
</p>projecteuclid.org/euclid.bjma/1513674118_20180108040158Mon, 08 Jan 2018 04:01 ESTWeighted Banach spaces of Lipschitz functionshttps://projecteuclid.org/euclid.bjma/1513674119<strong>A. Jiménez-Vargas</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 18 pages.</p><p><strong>Abstract:</strong><br/>
Given a pointed metric space $X$ and a weight $v$ on $\widetilde{X}$ (the complement of the diagonal set in $X\times X$ ), let $\mathrm{Lip}_{v}(X)$ and $\mathrm{lip}_{v}(X)$ denote the Banach spaces of all scalar-valued Lipschitz functions $f$ on $X$ vanishing at the basepoint such that $v\Phi(f)$ is bounded and $v\Phi(f)$ vanishes at infinity on $\widetilde{X}$ , respectively, where $\Phi(f)$ is the de Leeuw’s map of $f$ on $\widetilde{X}$ , under the weighted Lipschitz norm. The space $\mathrm{Lip}_{v}(X)$ has an isometric predual $\mathcal{F}_{v}(X)$ and it is proved that $(\mathrm{Lip}_{v}(X),\tau_{\operatorname{bw}^{*}})=(\mathcal{F}_{v}(X)^{*},\tau_{c})$ and $\mathcal{F}_{v}(X)=((\mathrm{Lip}_{v}(X),\tau_{\operatorname{bw}^{*}})',\tau_{c})$ , where $\tau_{\operatorname{bw}^{*}}$ denotes the bounded weak∗ topology and $\tau_{c}$ the topology of uniform convergence on compact sets. The linearization of the elements of $\mathrm{Lip}_{v}(X)$ is also tackled. Assuming that $X$ is compact, we address the question as to when $\mathrm{Lip}_{v}(X)$ is canonically isometrically isomorphic to $\mathrm{lip}_{v}(X)^{**}$ , and we show that this is the case whenever $\mathrm{lip}_{v}(X)$ is an M-ideal in $\mathrm{Lip}_{v}(X)$ and the so-called associated weights $\widetilde{v}_{L}$ and $\widetilde{v}_{l}$ coincide.
</p>projecteuclid.org/euclid.bjma/1513674119_20180108040158Mon, 08 Jan 2018 04:01 ESTGeneralized frames for operators associated with atomic systemshttps://projecteuclid.org/euclid.bjma/1512464418<strong>Dongwei Li</strong>, <strong>Jinsong Leng</strong>, <strong>Tingzhu Huang</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 16 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we investigate the g-frame and Bessel g-sequence related to a linear bounded operator $K$ in Hilbert space, which we call a $K$ -g-frame and a $K$ -dual Bessel g-sequence , respectively. Since the frame operator for a $K$ -g-frame may not be invertible, there is no classical canonical dual for a $K$ -g-frame. So we characterize the concept of a canonical $K$ -dual Bessel g-sequence of a $K$ -g-frame that generalizes the classical dual of a g-frame. Moreover, we use a family of linear operators to characterize atomic systems. We also consider the construction of new atomic systems from given ones and bounded operators.
</p>projecteuclid.org/euclid.bjma/1512464418_20180108040158Mon, 08 Jan 2018 04:01 ESTCalderón–Lozanovskii interpolation on quasi-Banach latticeshttps://projecteuclid.org/euclid.bjma/1512464419<strong>Yves Raynaud</strong>, <strong>Pedro Tradacete</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 20 pages.</p><p><strong>Abstract:</strong><br/>
We consider the Calderón–Lozanovskii construction $\varphi(X_{0},X_{1})$ in the context of quasi-Banach lattices, and we provide an extension of a result by Ovchinnikov concerning the associated interpolation methods $\varphi^{c}$ and $\varphi^{0}$ . Our approach is based on the interpolation properties of $(\infty,1)$ -regular operators between quasi-Banach lattices.
</p>projecteuclid.org/euclid.bjma/1512464419_20180108040158Mon, 08 Jan 2018 04:01 ESTStability of average roughness, octahedrality, and strong diameter $2$ properties of Banach spaces with respect to absolute sumshttps://projecteuclid.org/euclid.bjma/1512464420<strong>Rainis Haller</strong>, <strong>Johann Langemets</strong>, <strong>Rihhard Nadel</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 18 pages.</p><p><strong>Abstract:</strong><br/>
We prove that, if Banach spaces $X$ and $Y$ are $\delta$ -average rough, then their direct sum with respect to an absolute norm $N$ is $\delta/N(1,1)$ -average rough. In particular, for octahedral $X$ and $Y$ and for $p$ in $(1,\infty)$ , the space $X\oplus_{p}Y$ is $2^{1-1/p}$ -average rough, which is in general optimal. Another consequence is that for any $\delta$ in $(1,2]$ there is a Banach space which is exactly $\delta$ -average rough. We give a complete characterization when an absolute sum of two Banach spaces is octahedral or has the strong diameter 2 property. However, among all of the absolute sums, the diametral strong diameter 2 property is stable only for 1- and $\infty$ -sums.
</p>projecteuclid.org/euclid.bjma/1512464420_20180108040158Mon, 08 Jan 2018 04:01 ESTLocal matrix homotopies and soft torihttps://projecteuclid.org/euclid.bjma/1510909220<strong>Terry A. Loring</strong>, <strong>Fredy Vides</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 24 pages.</p><p><strong>Abstract:</strong><br/> We present solutions to local connectivity problems in matrix representations of the form $C([-1,1]^{N})\to C^{*}(u_{\varepsilon},v_{\varepsilon})$ , with $C_{\varepsilon}(\mathbb{T}^{2})\twoheadrightarrow C^{*}(u_{\varepsilon},v_{\varepsilon})$ for any $\varepsilon\in[0,2]$ and any integer $n\geq1$ , where $C^{*}(u_{\varepsilon},v_{\varepsilon})\subseteq M_{n}$ is an arbitrary matrix representation of the universal $C^{*}$ -algebra $C_{\varepsilon}(\mathbb{T}^{2})$ that denotes the soft torus . We solve the connectivity problems by introducing the so-called toroidal matrix links , which can be interpreted as normal contractive matrix analogies of free homotopies in differential algebraic topology. To deal with the locality constraints, we have combined some techniques introduced in this article with some techniques from matrix geometry, combinatorial optimization, and classification and representation theory of $C^{*}$ -algebras. </p>projecteuclid.org/euclid.bjma/1510909220_20180108040158Mon, 08 Jan 2018 04:01 ESTKolmogorov-type and general extension results for nonlinear expectationshttps://projecteuclid.org/euclid.bjma/1510909221<strong>Robert Denk</strong>, <strong>Michael Kupper</strong>, <strong>Max Nendel</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 26 pages.</p><p><strong>Abstract:</strong><br/>
We provide extension procedures for nonlinear expectations to the space of all bounded measurable functions. We first discuss a maximal extension for convex expectations which have a representation in terms of finitely additive measures. One of the main results of this article is an extension procedure for convex expectations which are continuous from above and therefore admit a representation in terms of countably additive measures. This can be seen as a nonlinear version of the Daniell–Stone theorem. From this, we deduce a robust Kolmogorov extension theorem which is then used to extend nonlinear kernels to an infinite-dimensional path space. We then apply this theorem to construct nonlinear Markov processes with a given family of nonlinear transition kernels.
</p>projecteuclid.org/euclid.bjma/1510909221_20180108040158Mon, 08 Jan 2018 04:01 ESTNew $L^{p}$ -inequalities for hyperbolic weights concerning the operators with complex Gaussian kernelshttps://projecteuclid.org/euclid.bjma/1510909222<strong>Benito J. González</strong>, <strong>Emilio R. Negrín</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 23 pages.</p><p><strong>Abstract:</strong><br/>
In this article the authors present a systematic study of several new $L^{p}$ -boundedness properties and Parseval-type relations concerning the operators with complex Gaussian kernels over the spaces $L^{p}(\mathbb{R},\cosh(\alpha x)\,dx)$ and $L^{p}(\mathbb{R},\cosh(\alpha x^{2})\,dx)$ , $1\leq p\leq\infty$ , $\alpha\in\mathbb{R}$ . Relevant connections with various earlier related results are also pointed out.
</p>projecteuclid.org/euclid.bjma/1510909222_20180108040158Mon, 08 Jan 2018 04:01 ESTNorm convergence of logarithmic means on unbounded Vilenkin groupshttps://projecteuclid.org/euclid.bjma/1510909224<strong>György Gát</strong>, <strong>Ushangi Goginava</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 17 pages.</p><p><strong>Abstract:</strong><br/>
In this paper we prove that, in the case of some unbounded Vilenkin groups, the Riesz logarithmic means converges in the norm of the spaces $X(G)$ for every $f\in X(G)$ , where by $X(G)$ we denote either the class of continuous functions with supremum norm or the class of integrable functions.
</p>projecteuclid.org/euclid.bjma/1510909224_20180108040158Mon, 08 Jan 2018 04:01 ESTCohomology for small categories: $k$ -graphs and groupoidshttps://projecteuclid.org/euclid.bjma/1510283128<strong>Elizabeth Gillaspy</strong>, <strong>Alexander Kumjian</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 28 pages.</p><p><strong>Abstract:</strong><br/>
Given a higher-rank graph $\Lambda$ , we investigate the relationship between the cohomology of $\Lambda$ and the cohomology of the associated groupoid $\mathcal{G}_{\Lambda}$ . We define an exact functor between the Abelian category of right modules over a higher-rank graph $\Lambda$ and the category of $\mathcal{G}_{\Lambda}$ -sheaves, where $\mathcal{G}_{\Lambda}$ is the path groupoid of $\Lambda$ . We use this functor to construct compatible homomorphisms from both the cohomology of $\Lambda$ with coefficients in a right $\Lambda$ -module, and the continuous cocycle cohomology of $\mathcal{G}_{\Lambda}$ with values in the corresponding $\mathcal{G}_{\Lambda}$ -sheaf, into the sheaf cohomology of $\mathcal{G}_{\Lambda}$ .
</p>projecteuclid.org/euclid.bjma/1510283128_20180108040158Mon, 08 Jan 2018 04:01 ESTVector lattices and $f$ -algebras: The classical inequalitieshttps://projecteuclid.org/euclid.bjma/1510283129<strong>G. Buskes</strong>, <strong>C. Schwanke</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 15 pages.</p><p><strong>Abstract:</strong><br/>
We present some of the classical inequalities in analysis in the context of Archimedean (real or complex) vector lattices and $f$ -algebras. In particular, we prove an identity for sesquilinear maps from the Cartesian square of a vector space to a geometric mean closed Archimedean vector lattice, from which a Cauchy–Schwarz inequality follows. A reformulation of this result for sesquilinear maps with a geometric mean closed semiprime Archimedean $f$ -algebra as codomain is also given. In addition, a sufficient and necessary condition for equality is presented. We also prove a Hölder inequality for weighted geometric mean closed Archimedean $\Phi$ -algebras, substantially improving results by K. Boulabiar and M. A. Toumi. As a consequence, a Minkowski inequality for weighted geometric mean closed Archimedean $\Phi$ -algebras is obtained.
</p>projecteuclid.org/euclid.bjma/1510283129_20180108040158Mon, 08 Jan 2018 04:01 ESTA variant of the Hankel multiplierhttps://projecteuclid.org/euclid.bjma/1510110961<strong>Saifallah Ghobber</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 23 pages.</p><p><strong>Abstract:</strong><br/>
The first aim of this article is to survey and revisit some uncertainty principles for the Hankel transform by means of the Hankel multiplier. Then we define the wavelet Hankel multiplier and study its boundedness and Schatten-class properties. Finally, we prove that the wavelet Hankel multiplier is unitary equivalent to a scalar multiple of the phase space restriction operator, for which we deduce a trace formula.
</p>projecteuclid.org/euclid.bjma/1510110961_20180108040158Mon, 08 Jan 2018 04:01 ESTToeplitz operators on weighted harmonic Bergman spaceshttps://projecteuclid.org/euclid.bjma/1510110962<strong>Zipeng Wang</strong>, <strong>Xianfeng Zhao</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 35 pages.</p><p><strong>Abstract:</strong><br/>
In this article, we study Toeplitz operators with nonnegative symbols on the $\mathcal{A}_{2}$ -weighted harmonic Bergman space. We characterize the boundedness, compactness, and invertibility of Toeplitz operators with nonnegative symbols on this space.
</p>projecteuclid.org/euclid.bjma/1510110962_20180108040158Mon, 08 Jan 2018 04:01 ESTOn the universal function for weighted spaces $L^{p}_{\mu}[0,1]$ , $p\geq 1$https://projecteuclid.org/euclid.bjma/1507017624<strong>Martin Grigoryan</strong>, <strong>Tigran Grigoryan</strong>, <strong>Artsrun Sargsyan</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 22 pages.</p><p><strong>Abstract:</strong><br/>
In this article, we show that there exist a function $g\in L^{1}[0,1]$ and a weight function $0\lt \mu(x)\leq1$ so that $g$ is universal for each class $L^{p}_{\mu}[0,1]$ , $p\geq 1$ , with respect to signs-subseries of its Fourier–Walsh series.
</p>projecteuclid.org/euclid.bjma/1507017624_20180108040158Mon, 08 Jan 2018 04:01 ESTHörmander-type theorems on unimodular multipliers and applications to modulation spaceshttps://projecteuclid.org/euclid.bjma/1507017625<strong>Qiang Huang</strong>, <strong>Jiecheng Chen</strong>, <strong>Dashan Fan</strong>, <strong>Xiangrong Zhu</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 19 pages.</p><p><strong>Abstract:</strong><br/>
In this article, for the unimodular multipliers $e^{i\mu (D)}$ , we establish two Hörmander-type multiplier theorems by assuming conditions on their phase functions $\mu $ . As applications, we obtain two multiplier theorems particularly fitting for the modulation spaces, thus allowing us to extend and improve some known results.
</p>projecteuclid.org/euclid.bjma/1507017625_20180108040158Mon, 08 Jan 2018 04:01 ESTApproximate uniqueness for maps from $C(X)$ into simple real rank $0$ C∗-algebrashttps://projecteuclid.org/euclid.bjma/1506412929<strong>P. W. Ng</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 18 pages.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a finite CW-complex, and let $\mathcal{A}$ be a unital separable simple finite $\mathcal{Z}$ -stable C∗-algebra with real rank $0$ . We prove an approximate uniqueness theorem for almost multiplicative contractive completely positive linear maps from $C(X)$ into $\mathcal{A}$ . We also give conditions for when such a map can, within a certain “error,” be approximated by a finite-dimensional ∗-homomorphism.
</p>projecteuclid.org/euclid.bjma/1506412929_20180108040158Mon, 08 Jan 2018 04:01 ESTDaugavet property and separability in Banach spaceshttps://projecteuclid.org/euclid.bjma/1505980813<strong>Abraham Rueda Zoca</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 17 pages.</p><p><strong>Abstract:</strong><br/>
We give a characterization of the separable Banach spaces with the Daugavet property which is applied to study the Daugavet property in the projective tensor product of an $L$ -embedded space with another nonzero Banach space. The former characterization also motivates the introduction and short study of two indices related to the Daugavet property.
</p>projecteuclid.org/euclid.bjma/1505980813_20180108040158Mon, 08 Jan 2018 04:01 ESTNearly relatively compact projections in operator algebrashttps://projecteuclid.org/euclid.bjma/1505959314<strong>Lawrence G. Brown</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 35 pages.</p><p><strong>Abstract:</strong><br/>
Let $A$ be a $C^{*}$ -algebra, and let $A^{**}$ be its enveloping von Neumann algebra. Akemann suggested a kind of noncommutative topology in which certain projections in $A^{**}$ play the role of open sets, and he used two operator inequalities in connection with compactness. Both of these inequalities are equivalent to compactness for a closed projection in $A^{**}$ , but only one is equivalent to relative compactness for a general projection. A third operator inequality, also related to compactness, was used by the author. The study of all three inequalities can be unified by considering a numerical invariant which is equivalent to the distance of a projection from the set of relatively compact projections. Tomita’s concept of regularity of projections seems relevant, and so we give some results and examples on regularity. We also include a few related results on semicontinuity.
</p>projecteuclid.org/euclid.bjma/1505959314_20180108040158Mon, 08 Jan 2018 04:01 ESTOn Banach spaces of vector-valued random variables and their duals motivated by risk measureshttps://projecteuclid.org/euclid.bjma/1504857611<strong>Thomas Kalmes</strong>, <strong>Alois Pichler</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 35 pages.</p><p><strong>Abstract:</strong><br/>
We introduce Banach spaces of vector-valued random variables motivated from mathematical finance. So-called risk functionals are defined in a natural way on these Banach spaces, and it is shown that these functionals are Lipschitz continuous. Since the risk functionals cannot be defined on strictly larger spaces of random variables, this creates an area of particular interest with regard to the spaces presented. We elaborate key properties of these Banach spaces and give representations of their dual spaces in terms of vector measures with values in the dual space of the state space.
</p>projecteuclid.org/euclid.bjma/1504857611_20180108040158Mon, 08 Jan 2018 04:01 ESTA generalized Hilbert operator acting on conformally invariant spaceshttps://projecteuclid.org/euclid.bjma/1504857614<strong>Daniel Girela</strong>, <strong>Noel Merchán</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 25 pages.</p><p><strong>Abstract:</strong><br/>
If $\mu$ is a positive Borel measure on the interval $[0,1)$ , we let $\mathcal{H}_{\mu}$ be the Hankel matrix $\mathcal{H}_{\mu}=(\mu_{n,k})_{n,k\ge0}$ with entries $\mu_{n,k}=\mu_{n+k}$ , where, for $n=0,1,2,\dots$ , $\mu_{n}$ denotes the moment of order $n$ of $\mu$ . This matrix formally induces the operator \[\mathcal{H}_{\mu}(f)(z)=\sum_{n=0}^{\infty}(\sum_{k=0}^{\infty}\mu_{n,k}{a_{k}})z^{n}\] on the space of all analytic functions $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ , in the unit disk $\mathbb{D}$ . This is a natural generalization of the classical Hilbert operator. The action of the operators $H_{\mu}$ on Hardy spaces has been recently studied. This article is devoted to a study of the operators $H_{\mu}$ acting on certain conformally invariant spaces of analytic functions on the disk such as the Bloch space, the space BMOA, the analytic Besov spaces, and the $Q_{s}$ -spaces.
</p>projecteuclid.org/euclid.bjma/1504857614_20180108040158Mon, 08 Jan 2018 04:01 ESTToeplitz operators on the space of real analytic functions: The Fredholm propertyhttps://projecteuclid.org/euclid.bjma/1501812082<strong>Pawel Domański</strong>, <strong>Michal Jasiczak</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 37 pages.</p><p><strong>Abstract:</strong><br/>
We completely characterize those continuous operators on the space of real analytic functions on the real line for which the associated matrix is Toeplitz (that is, we describe Toeplitz operators on this space). We also prove a necessary and sufficient condition for such operators to be Fredholm operators. While the space of real analytic functions is neither Banach space nor has a basis which makes available methods completely different from classical cases of Hardy spaces or Bergman spaces, nevertheless the results themselves show surprisingly strong similarity to the classical Hardy-space theory.
</p>projecteuclid.org/euclid.bjma/1501812082_20180108040158Mon, 08 Jan 2018 04:01 ESTNew function spaces related to Morrey spaces and the Fourier transformhttps://projecteuclid.org/euclid.bjma/1497600064<strong>Shohei Nakamura</strong>, <strong>Yoshihiro Sawano</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Advance publication, 30 pages.</p><p><strong>Abstract:</strong><br/>
We introduce new function spaces to handle the Fourier transform on Morrey spaces and investigate fundamental properties of the spaces. As an application, we generalize the Stein–Tomas Strichartz estimate to our spaces. The geometric property of Morrey spaces and related function spaces will improve some well-known estimates.
</p>projecteuclid.org/euclid.bjma/1497600064_20180108040158Mon, 08 Jan 2018 04:01 ESTToeplitz operators on weighted pluriharmonic Bergman spacehttps://projecteuclid.org/euclid.bjma/1516244456<strong>Linghui Kong</strong>, <strong>Yufeng Lu</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 2, 439--455.</p><p><strong>Abstract:</strong><br/>
In this article, we consider some algebraic properties of Toeplitz operators on weighted pluriharmonic Bergman space on the unit ball. We characterize the commutants of Toeplitz operators whose symbols are certain separately radial functions or holomorphic monomials, and then give a partial answer to the finite-rank product problem of Toeplitz operators.
</p>projecteuclid.org/euclid.bjma/1516244456_20180326040134Mon, 26 Mar 2018 04:01 EDTCompletely rank-nonincreasing multilinear mapshttps://projecteuclid.org/euclid.bjma/1520413214<strong>Hassan Yousefi</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 2, 481--496.</p><p><strong>Abstract:</strong><br/>
We extend the notion of completely rank-nonincreasing (CRNI) linear maps to include the multilinear maps. We show that a bilinear map on a finite-dimensional vector space on any field is CRNI if and only if it is a skew-compression bilinear map. We also characterize CRNI continuous bilinear maps defined on the set of compact operators.
</p>projecteuclid.org/euclid.bjma/1520413214_20180326040134Mon, 26 Mar 2018 04:01 EDTExtrapolation theorems for $(p,q)$ -factorable operatorshttps://projecteuclid.org/euclid.bjma/1520413213<strong>Orlando Galdames-Bravo</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 2, 497--514.</p><p><strong>Abstract:</strong><br/>
The operator ideal of $(p,q)$ -factorable operators can be characterized as the class of operators that factors through the embedding $L^{q'}(\mu)\hookrightarrow L^{p}(\mu)$ for a finite measure $\mu$ , where $p,q\in[1,\infty)$ are such that $1/p+1/q\ge1$ . We prove that this operator ideal is included into a Banach operator ideal characterized by means of factorizations through $r$ th and $s$ th power factorable operators, for suitable $r,s\in[1,\infty)$ . Thus, they also factor through a positive map $L^{s}(m_{1})^{\ast}\to L^{r}(m_{2})$ , where $m_{1}$ and $m_{2}$ are vector measures. We use the properties of the spaces of $u$ -integrable functions with respect to a vector measure and the $u$ th power factorable operators to obtain a characterization of $(p,q)$ -factorable operators and conditions under which a $(p,q)$ -factorable operator is $r$ -summing for $r\in[1,p]$ .
</p>projecteuclid.org/euclid.bjma/1520413213_20180326040134Mon, 26 Mar 2018 04:01 EDTComplex interpolation of predual spaces of general local Morrey-type spaceshttps://projecteuclid.org/euclid.bjma/1517281421<strong>Denny Ivanal Hakim</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 3, 541--571.</p><p><strong>Abstract:</strong><br/>
In this article, we investigate the complex interpolation of predual spaces of general local Morrey-type spaces. By showing that these spaces are equal to the associate space of general local Morrey-type spaces, we prove that predual spaces of general local Morrey-type spaces behave well under the first complex interpolation.
</p>projecteuclid.org/euclid.bjma/1517281421_20180703040033Tue, 03 Jul 2018 04:00 EDTOn a generalized uniform zero-two law for positive contractions of noncommutative $L_{1}$ -spaces and its vector-valued extensionhttps://projecteuclid.org/euclid.bjma/1525831240<strong>Inomjon Ganiev</strong>, <strong>Farrukh Mukhamedov</strong>, <strong>Dilmurod Bekbaev</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 3, 600--616.</p><p><strong>Abstract:</strong><br/>
Ornstein and Sucheston first proved that for a given positive contraction $T:L_{1}\to L_{1}$ there exists $m\in{\mathbb{N}}$ such that if $\Vert T^{m+1}-T^{m}\Vert \lt 2$ , then $\lim_{n\to\infty}\Vert T^{n+1}-T^{n}\Vert =0$ . This result was referred to as the zero-two law . In the present article, we prove a generalized uniform zero-two law for the multiparametric family of positive contractions of noncommutative $L_{1}$ -spaces. Moreover, we also establish a vector-valued analogue of the uniform zero-two law for positive contractions of $L_{1}(M,\Phi)$ —the noncommutative $L_{1}$ -spaces associated with center-valued traces.
</p>projecteuclid.org/euclid.bjma/1525831240_20180703040033Tue, 03 Jul 2018 04:00 EDTA generalized Schur complement for nonnegative operators on linear spaceshttps://projecteuclid.org/euclid.bjma/1524124812<strong>J. Friedrich</strong>, <strong>M. Günther</strong>, <strong>L. Klotz</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 3, 617--633.</p><p><strong>Abstract:</strong><br/>
Extending the corresponding notion for matrices or bounded linear operators on a Hilbert space, we define a generalized Schur complement for a nonnegative linear operator mapping a linear space into its dual, and we derive some of its properties.
</p>projecteuclid.org/euclid.bjma/1524124812_20180703040033Tue, 03 Jul 2018 04:00 EDTPointwise entangled ergodic theorems for Dunford–Schwartz operatorshttps://projecteuclid.org/euclid.bjma/1524124811<strong>Dávid Kunszenti-Kovács</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 3, 634--650.</p><p><strong>Abstract:</strong><br/>
We investigate pointwise convergence of entangled ergodic averages of Dunford–Schwartz operators $T_{0},T_{1},\ldots,T_{m}$ on a Borel probability space. These averages take the form
\[\frac{1}{N^{k}}\sum_{1\leq n_{1},\ldots,n_{k}\leq N}T_{m}^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\cdots A_{2}T_{2}^{n_{\alpha(2)}}A_{1}T_{1}^{n_{\alpha(1)}}f,\] where $f\in L^{p}(X,\mu)$ for some $1\leq p\lt \infty$ , and $\alpha:\{1,\ldots,m\}\to\{1,\ldots,k\}$ encodes the entanglement. We prove that, under some joint boundedness and twisted compactness conditions on the pairs $(A_{i},T_{i})$ , convergence holds almost everywhere for all $f\in L^{p}$ . We also present an extension to polynomial powers in the case $p=2$ , in addition to a continuous version concerning Dunford–Schwartz $C_{0}$ -semigroups.
</p>projecteuclid.org/euclid.bjma/1524124811_20180703040033Tue, 03 Jul 2018 04:00 EDTRotation of Gaussian paths on Wiener space with applicationshttps://projecteuclid.org/euclid.bjma/1517972422<strong>Seung Jun Chang</strong>, <strong>Jae Gil Choi</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 3, 651--672.</p><p><strong>Abstract:</strong><br/>
In this paper we first develop the rotation theorem of the Gaussian paths on Wiener space. We next analyze the generalized analytic Fourier–Feynman transform. As an application of our rotation theorem, we represent the multiple generalized analytic Fourier–Feynman transform as a single generalized Fourier–Feynman transform.
</p>projecteuclid.org/euclid.bjma/1517972422_20180703040033Tue, 03 Jul 2018 04:00 EDTSharp weighted bounds for fractional integrals via the two-weight theoryhttps://projecteuclid.org/euclid.bjma/1524124813<strong>Vakhtang Kokilashvili</strong>, <strong>Alexander Meskhi</strong>, <strong>Muhammad Asad Zaighum</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 3, 673--692.</p><p><strong>Abstract:</strong><br/>
We derive sharp weighted norm estimates for positive kernel operators on spaces of homogeneous type. Similar problems are studied for one-sided fractional integrals. Bounds of weighted norms are of mixed type. The problems are studied using the two-weight theory of positive kernel operators. As special cases, we derive sharp weighted estimates in terms of Muckenhoupt characteristics.
</p>projecteuclid.org/euclid.bjma/1524124813_20180703040033Tue, 03 Jul 2018 04:00 EDTSquare function inequalities for monotone bases in $L^{1}$https://projecteuclid.org/euclid.bjma/1529114493<strong>Adam Osękowski</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 3, 693--708.</p><p><strong>Abstract:</strong><br/>
We describe a novel method of handling general sharp square function inequalities for monotone bases and contractive projections in $L^{1}$ . The technique rests on the construction of an appropriate special function enjoying certain size and convexity-type properties. As an illustration, we establish a strong $L^{1}\to L^{1}$ and a weak-type $L^{1}\to L^{1,\infty}$ estimate for square functions.
</p>projecteuclid.org/euclid.bjma/1529114493_20180703040033Tue, 03 Jul 2018 04:00 EDTPerturbation analysis of the Moore–Penrose metric generalized inverse with applicationshttps://projecteuclid.org/euclid.bjma/1529114494<strong>Jianbing Cao</strong>, <strong>Yifeng Xue</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 12, Number 3, 709--729.</p><p><strong>Abstract:</strong><br/>
In this article, based on some geometric properties of Banach spaces and one feature of the metric projection, we introduce a new class of bounded linear operators satisfying the so-called $(\alpha,\beta)$ -USU ( uniformly strong uniqueness ) property . This new convenient property allows us to take the study of the stability problem of the Moore–Penrose metric generalized inverse a step further. As a result, we obtain various perturbation bounds of the Moore–Penrose metric generalized inverse of the perturbed operator. They offer the advantage that we do not need the quasiadditivity assumption, and the results obtained appear to be the most general case found to date. Closely connected to the main perturbation results, one application, the error estimate for projecting a point onto a linear manifold problem, is also investigated.
</p>projecteuclid.org/euclid.bjma/1529114494_20180703040033Tue, 03 Jul 2018 04:00 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 EDT