Banach Journal of Mathematical Analysis Articles (Project Euclid)
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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 EDTGeneralized $3$ -circular projections for unitary congruence invariant normshttp://projecteuclid.org/euclid.bjma/1463153910<strong>Abdullah Bin Abu Baker</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 3, 451--465.</p><p><strong>Abstract:</strong><br/>
A projection $P_{0}$ on a complex Banach space is generalized $3$ - circular if its linear combination with two projections $P_{1}$ and $P_{2}$ having coefficients $\lambda_{1}$ and $\lambda_{2}$ , respectively, is a surjective isometry, where $\lambda_{1}$ and $\lambda_{2}$ are distinct unit modulus complex numbers different from $1$ and $P_{0}\oplus P_{1}\oplus P_{2}=I$ . Such projections are always contractive. In this paper, we prove structure theorems for generalized $3$ -circular projections acting on the spaces of all $n\times n$ symmetric and skew-symmetric matrices over $\mathbb{C}$ when these spaces are equipped with unitary congruence invariant norms.
</p>projecteuclid.org/euclid.bjma/1463153910_20160513113834Fri, 13 May 2016 11:38 EDTCharacterizations of Jordan left derivations on some algebrashttp://projecteuclid.org/euclid.bjma/1463153911<strong>Guangyu An</strong>, <strong>Yana Ding</strong>, <strong>Jiankui Li</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 3, 466--481.</p><p><strong>Abstract:</strong><br/>
A linear mapping $\delta$ from an algebra $\mathcal{A}$ into a left $\mathcal{A}$ -module $\mathcal{M}$ is called a Jordan left derivation if $\delta(A^{2})=2A\delta(A)$ for every $A\in\mathcal{A}$ . We prove that if an algebra $\mathcal{A}$ and a left $\mathcal{A}$ -module $\mathcal{M}$ satisfy one of the following conditions—(1) $\mathcal{A}$ is a $C^{*}$ -algebra and $\mathcal{M}$ is a Banach left $\mathcal{A}$ -module; (2) $\mathcal{A}=\operatorname {Alg}\mathcal{L}$ with $\cap\{L_{-}:L\in\mathcal{J}_{\mathcal{L}}\}=(0)$ and $\mathcal{M}=B(X)$ ; and (3) $\mathcal{A}$ is a commutative subspace lattice algebra of a von Neumann algebra $\mathcal{B}$ and $\mathcal{M}=B(\mathcal{H})$ —then every Jordan left derivation from $\mathcal{A}$ into $\mathcal{M}$ is zero. $\delta$ is called left derivable at $G\in\mathcal{A}$ if $\delta(AB)=A\delta(B)+B\delta(A)$ for each $A,B\in\mathcal{A}$ with $AB=G$ . We show that if $\mathcal{A}$ is a factor von Neumann algebra, $G$ is a left separating point of $\mathcal{A}$ or a nonzero self-adjoint element in $\mathcal{A}$ , and $\delta$ is left derivable at $G$ , then $\delta\equiv0$ .
</p>projecteuclid.org/euclid.bjma/1463153911_20160513113834Fri, 13 May 2016 11:38 EDTOn certain uniformly open multilinear mappingshttp://projecteuclid.org/euclid.bjma/1463153912<strong>Marek Balcerzak</strong>, <strong>Ehrhard Behrends</strong>, <strong>Filip Strobin</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 3, 482--494.</p><p><strong>Abstract:</strong><br/>
We obtain two results stating the uniform openness of bilinear operators and multilinear functionals. The first result deals with Banach spaces $L^{p}:=L_{\mathbb{K}}^{p}$ (over $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$ ) and pointwise multiplication from $L^{p}\times L^{q}$ to $L^{r}$ (where $1/p+1/q=1/r$ ). The second result is concerned with the nontrivial $n$ -linear functionals from the product $X_{1}\times\cdots\times X_{n}$ of normed spaces (over $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$ ) to the field $\mathbb{K}$ .
</p>projecteuclid.org/euclid.bjma/1463153912_20160513113834Fri, 13 May 2016 11:38 EDTOn star, sharp, core, and minus partial orders in Rickart ringshttp://projecteuclid.org/euclid.bjma/1465230960<strong>Janko Marovt</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 3, 495--508.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{A}$ be a Rickart $\ast$ -ring and let $\leq^{\ast},\leq^{\sharp },\leq^{\oplus}$ , and $\leq_{\oplus}$ denote the star, the sharp, the core, and the dual core partial orders in $\mathcal{A}$ , respectively. The sets of all $b\in\mathcal{A}$ such that $a\leq b$ , along with the sets of all $b\in\mathcal{A}$ such that $b\leq a$ , are characterized, where $a\in\mathcal{A}$ is given and where $\leq $ is one of the partial orders: $\leq^{\ast}$ , or $\leq^{\sharp}$ , or $\leq^{\oplus}$ , or $\leq_{\oplus}$ . Such sets of elements that are above or below a given element under the minus partial order $\leq^{-}$ in a Rickart ring $\mathcal{A}$ are also studied. Some recent results of Cvetković-Ilić et al. on partial orders in $\mathcal{B}(H)$ , the algebra of all bounded linear operators on a Hilbert space $H$ , are thus generalized.
</p>projecteuclid.org/euclid.bjma/1465230960_20160606123605Mon, 06 Jun 2016 12:36 EDTDerivations on generalized semidirect products of Banach algebrashttp://projecteuclid.org/euclid.bjma/1465230961<strong>Hasan Pourmahmood Aghababa</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 3, 509--522.</p><p><strong>Abstract:</strong><br/>
Let $A$ and $B$ be Banach algebras, let $\theta:A\to B$ be a continuous Banach algebra homomorphism, and let $I$ be a closed ideal in $B$ . Then the $l^{1}$ -direct sum of $A$ and $I$ with a special product becomes a Banach algebra, denoted by $A\bowtie^{\theta}I$ , which we call the generalized semidirect product of $A$ and $I$ . In this article, among other things, we first characterize derivations on $A\bowtie^{\theta}I$ and then we compute the first cohomology group of $A\bowtie^{\theta}I$ when the first cohomology groups of $A$ with coefficients in $A$ and $I$ are trivial. As an application we characterize the first cohomology group of second duals of dual Banach algebras. Then we provide a nice characterization of the first cohomology group of $A\bowtie^{\mathrm{id}}A$ . Furthermore, we calculate the first cohomology group of some concrete Banach algebras related to locally compact groups.
</p>projecteuclid.org/euclid.bjma/1465230961_20160606123605Mon, 06 Jun 2016 12:36 EDTBoundary values of vector-valued Hardy spaces on nonsmooth domains and the Radon–Nikodym propertyhttp://projecteuclid.org/euclid.bjma/1465230962<strong>Hugo Ocampo-Salgado</strong>, <strong>Jorge Rivera-Noriega</strong>, <strong>Luis San Martin</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 3, 523--546.</p><p><strong>Abstract:</strong><br/>
We define Hardy spaces of functions taking values on a Banach space $\mathcal{X}$ over nonsmooth domains. The types of functions we consider are harmonic functions on a starlike Lipschitz domain and solutions to the heat equation on a time-varying domain. Our purpose is twofold: (a) to characterize the Radon–Nikodym property of the Banach space $\mathcal{X}$ in terms of the existence of nontangential limits of $\mathcal{X}$ -valued functions $u$ in the corresponding Hardy space with index $p\geq1$ , (b) to identify the function of the boundary values of $u$ in the Hardy space with index $p\gt 1$ with an element in the space $\mathcal{V}_{\mathcal{X}}^{p}$ of measures of $p$ -bounded variation in the absence of the Radon–Nikodym property of $\mathcal{X}$ . This extends similar results already known on the unit disk of $\mathbb{C}$ and the semispace $\mathbb{R}^{n}\times(0,\infty)$ .
</p>projecteuclid.org/euclid.bjma/1465230962_20160606123605Mon, 06 Jun 2016 12:36 EDTLinear maps between $\mathrm{C}^{*}$ -algebras preserving extreme points and strongly linear preservershttp://projecteuclid.org/euclid.bjma/1469199409<strong>María J. Burgos</strong>, <strong>Antonio C. Márquez-García</strong>, <strong>Antonio Morales-Campoy</strong>, <strong>Antonio M. Peralta</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 3, 547--565.</p><p><strong>Abstract:</strong><br/>
We study new classes of linear preservers between $\mathrm{C}^{*}$ -algebras and between $\mathrm{JB}^{*}$ -triples. Let $E$ and $F$ be $\mathrm{JB}^{*}$ -triples with $\partial_{e}(E_{1})\neq\emptyset$ . We prove that every linear map $T:E\to F$ strongly preserving Brown–Pedersen quasi-invertible elements is a triple homomorphism. Among the consequences, we establish that, given two unital $\mathrm{C}^{*}$ -algebras $A$ and $B$ , for each linear map $T$ strongly preserving Brown–Pedersen quasi-invertible elements, there exists a Jordan $^{*}$ -homomorphism $S:A\to B$ satisfying $T(x)=T(1)S(x)$ for every $x\in A$ . We also study the connections between linear maps strongly preserving Brown–Pedersen quasi-invertibility and other clases of linear preservers between $\mathrm{C}^{*}$ -algebras like Bergmann-zero pairs preservers, Brown–Pedersen quasi-invertibility preservers, and extreme points preservers.
</p>projecteuclid.org/euclid.bjma/1469199409_20160722105653Fri, 22 Jul 2016 10:56 EDTIntrinsic atomic and molecular decompositions of Hardy–Musielak–Orlicz spaceshttp://projecteuclid.org/euclid.bjma/1469199410<strong>Kwok-Pun Ho</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 3, 565--591.</p><p><strong>Abstract:</strong><br/>
We introduce the Hardy type space for Musielak–Orlicz spaces. It includes several existing Hardy type spaces such as the Hardy–Orlicz spaces and the Hardy spaces with variable exponents. Furthermore, we develop an atomic decomposition such that the size condition just relies on the norms of Musielak–Orlicz spaces. This gives us a nature extension of the molecular decompositions to the Hardy type space for Musielak–Orlicz spaces.
</p>projecteuclid.org/euclid.bjma/1469199410_20160722105653Fri, 22 Jul 2016 10:56 EDTSzegö-type decompositions for isometrieshttp://projecteuclid.org/euclid.bjma/1469199411<strong>Zbigniew Burdak</strong>, <strong>Marek Kosiek</strong>, <strong>Patryk Pagacz</strong>, <strong>Marek Słociński</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 3, 592--606.</p><p><strong>Abstract:</strong><br/>
The notion of Szegö-type properties of positive Borel measures is well known and widely exploited. In this paper, we consider a class of orthogonal decompositions of isometries on Hilbert spaces which correspond to Szegö-type properties of their elementary measures. Our decompositions are closely connected with some special families of invariant subspaces. It is shown that this connection holds for the decomposition constructed in the paper. We illustrate our results with several examples. We also give a short proof of Mlak’s theorem on the elementary measures of completely nonunitary contractions.
</p>projecteuclid.org/euclid.bjma/1469199411_20160722105653Fri, 22 Jul 2016 10:56 EDTVector-valued characters on vector-valued function algebrashttp://projecteuclid.org/euclid.bjma/1469199412<strong>Mortaza Abtahi</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 3, 607--619.</p><p><strong>Abstract:</strong><br/>
Let $A$ be a commutative unital Banach algebra and let $X$ be a compact space. We study the class of $A$ -valued function algebras on $X$ as subalgebras of $C(X,A)$ with certain properties. We introduce the notion of $A$ -characters of an $A$ -valued function algebra $\mathscr{A}$ as homomorphisms from $\mathscr{A}$ into $A$ that basically have the same properties as evaluation homomorphisms $\mathcal{E}_{x}:f\mapsto f(x)$ , with $x\in X$ . We show that, under certain conditions, there is a one-to-one correspondence between the set of $A$ -characters of $\mathscr{A}$ and the set of characters of the function algebra $\mathfrak{A}=\mathscr{A}\cap C(X)$ of all scalar-valued functions in $\mathscr{A}$ . For the so-called natural $A$ -valued function algebras , such as $C(X,A)$ and $\operatorname{Lip}(X,A)$ , we show that $\mathcal{E}_{x}$ ( $x\in X$ ) are the only $A$ -characters. Vector-valued characters are utilized to identify vector-valued spectra.
</p>projecteuclid.org/euclid.bjma/1469199412_20160722105653Fri, 22 Jul 2016 10:56 EDTNorm-attaining Lipschitz functionalshttp://projecteuclid.org/euclid.bjma/1471873728<strong>Vladimir Kadets</strong>, <strong>Miguel Martín</strong>, <strong>Mariia Soloviova</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 3, 621--637.</p><p><strong>Abstract:</strong><br/>
We prove that for a given Banach space $X$ , the subset of norm-attaining Lipschitz functionals in $\operatorname{Lip}_{0}(X)$ is weakly dense but not strongly dense. Then we introduce a weaker concept of directional norm attainment and demonstrate that for a uniformly convex $X$ the set of directionally norm-attaining Lipschitz functionals is strongly dense in $\operatorname{Lip}_{0}(X)$ and, moreover, that an analogue of the Bishop–Phelps–Bollobás theorem is valid.
</p>projecteuclid.org/euclid.bjma/1471873728_20160822094855Mon, 22 Aug 2016 09:48 EDTEquations for frame wavelets in $L^{2}(\mathbb{R}^{2})$http://projecteuclid.org/euclid.bjma/1471873729<strong>Xingde Dai</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 3, 638--670.</p><p><strong>Abstract:</strong><br/>
A finite solution to a system of equations will generate a single function normalized tight frame wavelet (Parseval’s frame wavelet) with compact support associated with a $2\times2$ expansive integral matrix whose determinant is either $2$ or $-2$ in $L^{2}(\mathbb{R}^{2})$ .
</p>projecteuclid.org/euclid.bjma/1471873729_20160822094855Mon, 22 Aug 2016 09:48 EDTAn extension of a theorem of Schoenberg to products of sphereshttp://projecteuclid.org/euclid.bjma/1472657851<strong>J. C. Guella</strong>, <strong>V. A. Menegatto</strong>, <strong>A. P. Peron</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 4, 671--685.</p><p><strong>Abstract:</strong><br/>
We present a characterization for the continuous, isotropic, and positive definite kernels on a product of spheres along the lines of a classical result of Schoenberg on positive definiteness on a single sphere. We also discuss a few issues regarding the characterization, including topics for future investigation.
</p>projecteuclid.org/euclid.bjma/1472657851_20160831113736Wed, 31 Aug 2016 11:37 EDTIdeal structures in vector-valued polynomial spaceshttp://projecteuclid.org/euclid.bjma/1472657852<strong>Verónica Dimant</strong>, <strong>Silvia Lassalle</strong>, <strong>Ángeles Prieto</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 4, 686--702.</p><p><strong>Abstract:</strong><br/>
This paper is concerned with the study of geometric structures in spaces of polynomials. More precisely, we discuss for $E$ and $F$ Banach spaces, whether the class of $n$ -homogeneous polynomials, $\mathcal{P}_{w}(^{n}E,F)$ , which are weakly continuous on bounded sets, is an HB-subspace or an $M(1,C)$ -ideal in the space of continuous $n$ -homogeneous polynomials, $\mathcal{P}(^{n}E,F)$ . We establish sufficient conditions under which the problem can be positively solved. Some examples are given. We also study when some ideal structures pass from $\mathcal{P}_{w}(^{n}E,F)$ as an ideal in $\mathcal{P}(^{n}E,F)$ to the range space $F$ as an ideal in its bidual $F^{**}$ .
</p>projecteuclid.org/euclid.bjma/1472657852_20160831113736Wed, 31 Aug 2016 11:37 EDTNoncommutative Hardy–Lorentz spaces associated with semifinite subdiagonal algebrashttp://projecteuclid.org/euclid.bjma/1472657853<strong>Yazhou Han</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 4, 703--726.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{A}$ be a maximal subdiagonal algebra of semifinite von Neumann algebra $\mathcal{M}$ . For $0\lt p\leq\infty$ , we define the noncommutative Hardy–Lorentz spaces $H^{p,\omega}(\mathcal{A})$ , and give some properties of these spaces. We obtain that the Herglotz maps are bounded linear maps from $\Lambda_{\omega}^{p}(\mathcal{M})$ into $\Lambda_{\omega}^{p}(\mathcal{M})$ , and with this result we characterize the dual spaces of $H^{p,\omega}(\mathcal{A})$ for $1\lt p\lt \infty$ . We also present the Hartman–Wintner spectral inclusion theorem of Toeplitz operators and Pisier’s interpolation theorem for this case.
</p>projecteuclid.org/euclid.bjma/1472657853_20160831113736Wed, 31 Aug 2016 11:37 EDTPoisson semigroup, area function, and the characterization of Hardy space associated to degenerate Schrödinger operatorshttp://projecteuclid.org/euclid.bjma/1472657854<strong>Jizheng Huang</strong>, <strong>Pengtao Li</strong>, <strong>Yu Liu</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 4, 727--749.</p><p><strong>Abstract:</strong><br/>
Let
\begin{eqnarray*}Lf(x)=-\frac{1}{\omega(x)}\sum_{i,j}\partial_{i}(a_{ij}(\cdot)\partial _{j}f)(x)+V(x)f(x)\end{eqnarray*} be the degenerate Schrödinger operator, where $\omega$ is a weight from the Muckenhoupt class $A_{2}$ and $V$ is a nonnegative potential that belongs to a certain reverse Hölder class with respect to the measure $\omega(x)dx$ . Based on some smoothness estimates of the Poisson semigroup $e^{-t\sqrt{L}}$ , we introduce the area function $S^{L}_{P}$ associated with $e^{-t\sqrt{L}}$ to characterize the Hardy space associated with $L$ .
</p>projecteuclid.org/euclid.bjma/1472657854_20160831113736Wed, 31 Aug 2016 11:37 EDTMartingale Hardy spaces with variable exponentshttp://projecteuclid.org/euclid.bjma/1474373751<strong>Yong Jiao</strong>, <strong>Dejian Zhou</strong>, <strong>Zhiwei Hao</strong>, <strong>Wei Chen</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 4, 750--770.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce Hardy spaces with variable exponents defined on a probability space and develop the martingale theory of variable Hardy spaces. We prove the weak-type and strong-type inequalities on Doob’s maximal operator, and we get a $(1,p(\cdot),\infty)$ -atomic decomposition for Hardy martingale spaces associated with conditional square functions. As applications, we obtain a dual theorem and the John–Nirenberg inequalities in the frame of variable exponents. The key ingredient is that we find a condition with a probabilistic characterization of $p(\cdot)$ to replace the so-called log-Hölder continuity condition in $\mathbb{R}^{n}$ .
</p>projecteuclid.org/euclid.bjma/1474373751_20160920081558Tue, 20 Sep 2016 08:15 EDTSubspaces of Banach spaces with big sliceshttp://projecteuclid.org/euclid.bjma/1474373752<strong>Julio Becerra Guerrero</strong>, <strong>Ginés López-Pérez</strong>, <strong>Abraham Rueda Zoca</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 4, 771--782.</p><p><strong>Abstract:</strong><br/>
We study when diameter $2$ properties can be inherited by subspaces. We obtain that the slice diameter $2$ property (resp., the diameter $2$ property, strong diameter $2$ property) passes from a Banach space $X$ to a subspace $Y$ whenever $X/Y$ is finite-dimensional and $Y$ is complemented by a norm $1$ projection (resp., the quotient $X/Y$ is finite-dimensional and strongly regular). Also, we study the same problem for the dual properties of diameter $2$ properties, such as having octahedral, weakly octahedral, or $2$ -rough norm.
</p>projecteuclid.org/euclid.bjma/1474373752_20160920081558Tue, 20 Sep 2016 08:15 EDTCover-strict topologies, ideals, and quotients for some spaces of vector-valued functionshttp://projecteuclid.org/euclid.bjma/1474373753<strong>Terje Hõim</strong>, <strong>D. A. Robbins</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 4, 783--799.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a completely regular Hausdorff space, let $\mathcal{D}$ be a cover of $X$ , and let $\pi:\mathcal{E}\rightarrow X$ be a bundle of Banach spaces (algebras). Let $\Gamma (\pi)$ be the space of sections of $\pi $ , and let $\Gamma _{b}(\pi,\mathcal{D})$ be the subspace of $\Gamma(\pi)$ consisting of sections which are bounded on each $D\in \mathcal{D}$ . We study the subspace (ideal) and quotient structures of some spaces of vector-valued functions which arise from endowing $\Gamma _{b}(\pi,\mathcal{D})$ with the cover-strict topology.
</p>projecteuclid.org/euclid.bjma/1474373753_20160920081558Tue, 20 Sep 2016 08:15 EDTInequalities on the spectral radius and the operator norm of Hadamard products of positive operators on sequence spaceshttp://projecteuclid.org/euclid.bjma/1474373754<strong>Roman Drnovšek</strong>, <strong>Aljoša Peperko</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 4, 800--814.</p><p><strong>Abstract:</strong><br/>
K. M. R. Audenaert (2010), R. A. Horn and F. Zhang (2010), Z. Huang (2011), A. R. Schep (2011), A. Peperko (2012), and D. Chen and Y. Zhang (2015) have proved inequalities on the spectral radius and the operator norm of Hadamard products and ordinary matrix products of finite and infinite nonnegative matrices that define operators on sequence spaces. In the present article, we extend and refine several of these results, and we also prove some analogues for the numerical radius.
</p>projecteuclid.org/euclid.bjma/1474373754_20160920081558Tue, 20 Sep 2016 08:15 EDTInterpolation with a parameter function of $L^{p}$ -spaces with respect to a vector measure on a $\delta$ -ringhttp://projecteuclid.org/euclid.bjma/1475267149<strong>R. del Campo</strong>, <strong>A. Fernández</strong>, <strong>A. Manzano</strong>, <strong>F. Mayoral</strong>, <strong>F. Naranjo</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 4, 815--827.</p><p><strong>Abstract:</strong><br/>
Let $\nu$ be a $\sigma$ -finite Banach-space-valued measure defined on a $\delta$ -ring. We find a wide class of measures $\nu$ for which interpolation with a parameter function of couples of Banach lattices of $p$ -integrable and weakly $p$ -integrable functions with respect to $\nu$ produces a Lorentz-type space. Moreover, we prove that if we interpolate between sums and intersections of them, then they still yield another Lorentz-type space closely related with the first one.
</p>projecteuclid.org/euclid.bjma/1475267149_20160930162554Fri, 30 Sep 2016 16:25 EDTOn the stability of the orthogonality equation and the orthogonality-preserving property with two unknown functionshttp://projecteuclid.org/euclid.bjma/1475267150<strong>Jacek Chmieliński</strong>, <strong>Radosław Łukasik</strong>, <strong>Paweł Wójcik</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 4, 828--847.</p><p><strong>Abstract:</strong><br/>
For two unknown functions $f$ , $g$ , the equation
\[\langle f(x)|g(y)\rangle=\langle x\mid y\rangle\] and its stability as well as the approximate orthogonality-preserving property \[x\perp y\Longrightarrow fx\perp^{\varepsilon}gy\] are considered.
</p>projecteuclid.org/euclid.bjma/1475267150_20160930162554Fri, 30 Sep 2016 16:25 EDTWandering subspaces and fusion frame generators for unitary systemshttp://projecteuclid.org/euclid.bjma/1475870135<strong>Aifang Liu</strong>, <strong>Pengtong Li</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 4, 848--863.</p><p><strong>Abstract:</strong><br/>
This work is inspired by the study of wandering vectors and frame vectors for unitary systems. We investigate the structure and properties of complete wandering subspaces for unitary systems, and, in particular, we consider the unitary systems with a structure similar to wavelet systems. Given a unitary system with a complete wandering subspace, a necessary and sufficient condition for a closed subspace to be a Parseval fusion frame generator is obtained. Moreover, we study the dilation property for Parseval fusion frame generators for unitary groups.
</p>projecteuclid.org/euclid.bjma/1475870135_20161007155543Fri, 07 Oct 2016 15:55 EDTConvex cones of generalized multiply monotone functions and the dual coneshttp://projecteuclid.org/euclid.bjma/1475870136<strong>Iosif Pinelis</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 10, Number 4, 864--897.</p><p><strong>Abstract:</strong><br/>
Let $n$ and $k$ be nonnegative integers such that $1\le k\leq n+1$ . The convex cone $\mathscr{F}_{+}^{k:n}$ of all functions $f$ on an arbitrary interval $I\subseteq\mathbb{R}$ whose derivatives $f^{(j)}$ of orders $j=k-1,\dots,n$ are nondecreasing is characterized. A simple description of the convex cone dual to $\mathscr{F}_{+}^{k:n}$ is given. In particular, these results are useful in, and were motivated by, applications in probability. In fact, the results are obtained in a more general setting with certain generalized derivatives of $f$ of the $j$ th order in place of $f^{(j)}$ . Somewhat similar results were previously obtained, in terms of Tchebycheff–Markov systems, in the case when the left endpoint of the interval $I$ is finite, with certain additional integrability conditions; such conditions fail to hold in the mentioned applications. Development of substantially new methods was needed to overcome the difficulties.
</p>projecteuclid.org/euclid.bjma/1475870136_20161007155543Fri, 07 Oct 2016 15:55 EDTFrames and representing systems in Fréchet spaces and their dualshttp://projecteuclid.org/euclid.bjma/1476841711<strong>J. Bonet</strong>, <strong>C. Fernández</strong>, <strong>A. Galbis</strong>, <strong>J. M. Ribera</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 1, 1--20.</p><p><strong>Abstract:</strong><br/>
Frames and Bessel sequences in Fréchet spaces and their duals are defined and studied. Their relation with Schauder frames and representing systems is analyzed. The abstract results presented here, when applied to concrete spaces of analytic functions, give examples and consequences about sampling sets and Dirichlet series expansions.
</p>projecteuclid.org/euclid.bjma/1476841711_20161018214836Tue, 18 Oct 2016 21:48 EDTHarmonic analysis on the Proper Velocity gyrogrouphttp://projecteuclid.org/euclid.bjma/1476841712<strong>Milton Ferreira</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 1, 21--49.</p><p><strong>Abstract:</strong><br/>
In this article we study harmonic analysis on the proper velocity (PV) gyrogroup using the gyrolanguage of analytic hyperbolic geometry. This PV addition is the relativistic addition of proper velocities in special relativity, and it is related with the hyperboloid model of hyperbolic geometry. The generalized harmonic analysis depends on a complex parameter $z$ and on the radius $t$ of the hyperboloid, and it comprises the study of the generalized translation operator, the associated convolution operator, the generalized Laplace–Beltrami operator and its eigenfunctions, the generalized Poisson transform and its inverse, the generalized Helgason–Fourier transform and its inverse, and Plancherel’s theorem. In the limit of large $t$ , $t\rightarrow+\infty$ , the generalized harmonic analysis on the hyperboloid tends to the standard Euclidean harmonic analysis on ${\mathbb{R}}^{n}$ , thus unifying hyperbolic and Euclidean harmonic analysis.
</p>projecteuclid.org/euclid.bjma/1476841712_20161018214836Tue, 18 Oct 2016 21:48 EDTAbstract harmonic analysis of wave-packet transforms over locally compact abelian groupshttp://projecteuclid.org/euclid.bjma/1478746986<strong>Arash Ghaani Farashahi</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 1, 50--71.</p><p><strong>Abstract:</strong><br/>
This article presents a systematic study for abstract harmonic analysis aspects of wave-packet transforms over locally compact abelian (LCA) groups. Let $H$ be a locally compact group, let $K$ be an LCA group, and let $\theta:H\to\operatorname{Aut}(K)$ be a continuous homomorphism. We introduce the abstract notion of the wave-packet group generated by $\theta$ , and we study basic properties of wave-packet groups. Then we study theoretical aspects of wave-packet transforms. Finally, we will illustrate application of these techniques in the case of some well-known examples.
</p>projecteuclid.org/euclid.bjma/1478746986_20161109220314Wed, 09 Nov 2016 22:03 ESTDuality for increasing convex functionals with countably many marginal constraintshttp://projecteuclid.org/euclid.bjma/1478746987<strong>Daniel Bartl</strong>, <strong>Patrick Cheridito</strong>, <strong>Michael Kupper</strong>, <strong>Ludovic Tangpi</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 1, 72--89.</p><p><strong>Abstract:</strong><br/>
In this work we derive a convex dual representation for increasing convex functionals on a space of real-valued Borel measurable functions defined on a countable product of metric spaces. Our main assumption is that the functionals fulfill marginal constraints satisfying a certain tightness condition. In the special case where the marginal constraints are given by expectations or maxima of expectations, we obtain linear and sublinear versions of Kantorovich’s transport duality and the recently discovered martingale transport duality on products of countably many metric spaces.
</p>projecteuclid.org/euclid.bjma/1478746987_20161109220314Wed, 09 Nov 2016 22:03 ESTSpaceability in norm-attaining setshttp://projecteuclid.org/euclid.bjma/1478746988<strong>Javier Falcó</strong>, <strong>Domingo García</strong>, <strong>Manuel Maestre</strong>, <strong>Pilar Rueda</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 1, 90--107.</p><p><strong>Abstract:</strong><br/>
We study the existence of infinite-dimensional vector spaces in the sets of norm-attaining operators, multilinear forms, and polynomials. Our main result is that, for every set of permutations $P$ of the set $\{1,\ldots,n\}$ , there exists a closed infinite-dimensional Banach subspace of the space of $n$ -linear forms on $\ell_{1}$ such that, for all nonzero elements $B$ of such a subspace, the Arens extension associated to the permutation $\sigma$ of $B$ is norm-attaining if and only if $\sigma$ is an element of $P$ . We also study the structure of the set of norm-attaining $n$ -linear forms on $c_{0}$ .
</p>projecteuclid.org/euclid.bjma/1478746988_20161109220314Wed, 09 Nov 2016 22:03 ESTDuality for ideals of Lipschitz mapshttp://projecteuclid.org/euclid.bjma/1478746989<strong>M. G. Cabrera-Padilla</strong>, <strong>J. A. Chávez-Domínguez</strong>, <strong>A. Jiménez-Vargas</strong>, <strong>Moisés Villegas-Vallecillos</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 1, 108--129.</p><p><strong>Abstract:</strong><br/>
We develop a systematic approach to the study of ideals of Lipschitz maps from a metric space to a Banach space, inspired by classical theory on using Lipschitz tensor products to relate ideals of operator/tensor norms for Banach spaces. We study spaces of Lipschitz maps from a metric space to a dual Banach space that can be represented canonically as the dual of a Lipschitz tensor product endowed with a Lipschitz cross-norm, and we show that several known examples of ideals of Lipschitz maps (Lipschitz maps, Lipschitz $p$ -summing maps, maps admitting Lipschitz factorization through subsets of $L_{p}$ -space) admit such a representation. Generally, we characterize when the space of a Lipschitz map from a metric space to a dual Banach space is in canonical duality with a Lipschitz cross-norm. Finally, we introduce a concept of operators which are approximable with respect to one of these ideals of Lipschitz maps, and we identify them in terms of tensor-product notions.
</p>projecteuclid.org/euclid.bjma/1478746989_20161109220314Wed, 09 Nov 2016 22:03 ESTHlawka’s functional inequality on topological groupshttp://projecteuclid.org/euclid.bjma/1480474816<strong>Włodzimierz Fechner</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 1, 130--142.</p><p><strong>Abstract:</strong><br/>
Let $(X,+)$ be a topological abelian group. We discuss regularity of solutions $f\colon X\to\mathbb{R}$ of Hlawka’s functional inequality
\[f(x+y)+f(y+z)+f(x+z)\leq f(x+y+z)+f(x)+f(y)+f(z),\] postulated for all $x,y,z\in X$ . We study the lower and upper hull of $f$ . Moreover, we provide conditions which imply continuity of $f$ . We prove, in particular, that if $X$ is generated by any neighborhood of zero, $f$ is continuous at zero, and $f(0)=0$ , then $f$ is continuous on $X$ .
</p>projecteuclid.org/euclid.bjma/1480474816_20161129220026Tue, 29 Nov 2016 22:00 ESTApproximative compactness in Musielak–Orlicz function spaces of Bochner typehttp://projecteuclid.org/euclid.bjma/1480474817<strong>Shaoqiang Shang</strong>, <strong>Yunan Cui</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 1, 143--163.</p><p><strong>Abstract:</strong><br/>
In this article, we give the criteria for approximative compactness of every proximinal convex subset of Musielak–Orlicz–Bochner function spaces equipped with the Orlicz norm. As a corollary, we give the criteria for approximative compactness of Musielak–Orlicz–Bochner function spaces equipped with the Orlicz norm.
</p>projecteuclid.org/euclid.bjma/1480474817_20161129220026Tue, 29 Nov 2016 22:00 ESTUnbounded composition operators via inductive limits: Cosubnormal operators with matrix symbols, IIhttp://projecteuclid.org/euclid.bjma/1480474818<strong>Piotr Budzyński</strong>, <strong>Piotr Dymek</strong>, <strong>Artur Płaneta</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 1, 164--187.</p><p><strong>Abstract:</strong><br/>
This article deals with unbounded composition operators with infinite matrix symbols acting in $L^{2}$ -spaces with respect to the Gaussian measure on $\mathbb{R}^{\infty}$ . We introduce weak cohyponormality classes $\mathcal{S}_{n,r}^{*}$ of unbounded operators and provide criteria for the aforementioned composition operators to belong to $\mathcal{S}_{n,r}^{*}$ . Our approach is based on inductive limits of operators.
</p>projecteuclid.org/euclid.bjma/1480474818_20161129220026Tue, 29 Nov 2016 22:00 EST$\ell_{p}$ -maximal regularity for a class of fractional difference equations on UMD spaces: The case $1\lt \alpha\leq2$http://projecteuclid.org/euclid.bjma/1480474819<strong>Carlos Lizama</strong>, <strong>Marina Murillo-Arcila</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 1, 188--206.</p><p><strong>Abstract:</strong><br/>
By using Blunck’s operator-valued Fourier multiplier theorem, we completely characterize the existence and uniqueness of solutions in Lebesgue sequence spaces for a discrete version of the Cauchy problem with fractional order $1\lt \alpha\leq2$ . This characterization is given solely in spectral terms on the data of the problem, whenever the underlying Banach space belongs to the UMD-class.
</p>projecteuclid.org/euclid.bjma/1480474819_20161129220026Tue, 29 Nov 2016 22:00 ESTOrder structure, multipliers, and Gelfand representation of vector-valued function algebrashttp://projecteuclid.org/euclid.bjma/1481274115<strong>Jorma Arhippainen</strong>, <strong>Jukka Kauppi</strong>, <strong>Jussi Mattas</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 1, 207--222.</p><p><strong>Abstract:</strong><br/>
We continue the study begun by the third author of $C^{*}$ -Segal algebra-valued function algebras with an emphasis on the order structure. Our main result is a characterization theorem for $C^{*}$ -Segal algebra-valued function algebras with an order unitization. As an intermediate step, we establish a function algebraic description of the multiplier module of arbitrary Segal algebra-valued function algebras. We also consider the Gelfand representation of these algebras in the commutative case.
</p>projecteuclid.org/euclid.bjma/1481274115_20161209040212Fri, 09 Dec 2016 04:02 ESTTriangular summability and Lebesgue points of $2$ -dimensional Fourier transformshttp://projecteuclid.org/euclid.bjma/1481274116<strong>Ferenc Weisz</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 1, 223--238.</p><p><strong>Abstract:</strong><br/>
We consider the triangular $\theta$ -summability of $2$ -dimensional Fourier transforms. Under some conditions on $\theta$ , we show that the triangular $\theta$ -means of a function $f$ belonging to the Wiener amalgam space $W(L_{1},\ell_{\infty})({\mathbb{R}}^{2})$ converge to $f$ at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points for the so-called modified Lebesgue points of $f\in W(L_{p},\ell_{\infty})({\mathbb{R}}^{2})$ whenever $1\lt p\lt \infty$ . Some special cases of the $\theta$ -summation are considered, such as the Weierstrass, Abel, Picard, Bessel, Fejér, de La Vallée-Poussin, Rogosinski, and Riesz summations.
</p>projecteuclid.org/euclid.bjma/1481274116_20161209040212Fri, 09 Dec 2016 04:02 ESTErgodic behaviors of the regular operator meanshttp://projecteuclid.org/euclid.bjma/1484363107<strong>Laurian Suciu</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 2, 239--265.</p><p><strong>Abstract:</strong><br/>
This article deals with some ergodic properties for general sequences in the closed convex hull of the orbit of some (not necessarily power-bounded) operators in Banach spaces. A regularity condition more general than that of ergodicity is used to obtain some versions of the Esterle–Katznelson–Tzafriri theorem. Also, the ergodicity of the backward iterates of a sequence is proved under appropriate assumptions as, for example, its peripheral boundedness on the unit circle. The applications concern uniformly Kreiss-bounded operators, and other ergodic results are obtained for the binomial means and some operator means related to the Cesàro means.
</p>projecteuclid.org/euclid.bjma/1484363107_20170113220512Fri, 13 Jan 2017 22:05 ESTExtended spectrum and extended eigenspaces of quasinormal operatorshttp://projecteuclid.org/euclid.bjma/1484363108<strong>Gilles Cassier</strong>, <strong>Hasan Alkanjo</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 2, 266--281.</p><p><strong>Abstract:</strong><br/>
We say that a complex number $\lambda$ is an extended eigenvalue of a bounded linear operator $T$ on a Hilbert space $\mathcal{H}$ if there exists a nonzero bounded linear operator $X$ acting on $\mathcal{H}$ , called the extended eigenvector associated to $\lambda$ , and satisfying the equation $TX=\lambda XT$ . In this article, we describe the sets of extended eigenvalues and extended eigenvectors for the quasinormal operators.
</p>projecteuclid.org/euclid.bjma/1484363108_20170113220512Fri, 13 Jan 2017 22:05 ESTOn Cohomology for Product Systemshttp://projecteuclid.org/euclid.bjma/1484816415<strong>Jeong Hee Hong</strong>, <strong>Mi Jung Son</strong>, <strong>Wojciech Szymański</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 2, 282--294.</p><p><strong>Abstract:</strong><br/>
A cohomology for product systems of Hilbert bimodules is defined via the $\operatorname{Ext}$ functor. For the class of product systems corresponding to irreversible algebraic dynamics, relevant resolutions are found explicitly and it is shown how the underlying product system can be twisted by 2-cocycles. In particular, this process gives rise to cohomological deformations of the $C^{*}$ -algebras associated with the product system. Concrete examples of deformations of the Cuntz’s algebra ${\mathcal{Q}}_{\mathbb{N}}$ arising this way are investigated, and we show that they are simple and purely infinite.
</p>projecteuclid.org/euclid.bjma/1484816415_20170119040053Thu, 19 Jan 2017 04:00 ESTThe approximate hyperplane series property and related propertieshttp://projecteuclid.org/euclid.bjma/1484816416<strong>María D. Acosta</strong>, <strong>Richard Martin Aron</strong>, <strong>Francisco Javier García-Pacheco</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 2, 295--310.</p><p><strong>Abstract:</strong><br/>
We show that the approximate hyperplane series property consequence, we obtain that the class of spaces $Y$ such that the pair $(\ell_{1},Y)$ has the Bishop–Phelps–Bollobás property for operators is stable under finite $\ell_{p}$ -sums for $1\leq p\lt \infty$ . We also deduce that every Banach space of dimension at least $2$ can be equivalently renormed to have the AHSp but to fail Lindenstrauss’ property $\beta$ . We also show that every infinite-dimensional Banach space admitting an equivalent strictly convex norm also admits such an equivalent norm failing the AHSp.
</p>projecteuclid.org/euclid.bjma/1484816416_20170119040053Thu, 19 Jan 2017 04:00 ESTComposition operators on the Bloch space of the unit ball of a Hilbert spacehttp://projecteuclid.org/euclid.bjma/1485572420<strong>Oscar Blasco</strong>, <strong>Pablo Galindo</strong>, <strong>Mikael Lindström</strong>, <strong>Alejandro Miralles</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 2, 311--334.</p><p><strong>Abstract:</strong><br/>
Every analytic self-map of the unit ball of a Hilbert space induces a bounded composition operator on the space of Bloch functions. Necessary and sufficient conditions for compactness of such composition operators are provided, as well as some examples that clarify the connections among such conditions.
</p>projecteuclid.org/euclid.bjma/1485572420_20170127220034Fri, 27 Jan 2017 22:00 ESTNorm estimates for random polynomials on the scale of Orlicz spaceshttp://projecteuclid.org/euclid.bjma/1485572421<strong>Andreas Defant</strong>, <strong>Mieczysław Mastyło</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 2, 335--347.</p><p><strong>Abstract:</strong><br/>
We prove an upper bound for the supremum norm of homogeneous Bernoulli polynomials on the unit ball of finite-dimensional complex Banach spaces. This result is inspired by the famous Kahane–Salem–Zygmund inequality and its recent extensions; in contrast to the known results, our estimates are on the scale of Orlicz spaces instead of $\ell_{p}$ -spaces. Applications are given to multidimensional Bohr radii for holomorphic functions in several complex variables, and to the study of unconditionality of spaces of homogenous polynomials in Banach spaces.
</p>projecteuclid.org/euclid.bjma/1485572421_20170127220034Fri, 27 Jan 2017 22:00 ESTNew results on Kottman’s constanthttp://projecteuclid.org/euclid.bjma/1487732418<strong>Jesús M. F. Castillo</strong>, <strong>Manuel González</strong>, <strong>Pier Luigi Papini</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 2, 348--362.</p><p><strong>Abstract:</strong><br/>
We present new results on Kottman’s constant of a Banach space, showing (i) that every Banach space is isometric to a hyperplane of a Banach space having Kottman’s constant 2 and (ii) that Kottman’s constant of a Banach space and of its bidual can be different. We say that a Banach space is a Diestel space if the infimum of Kottman’s constants of its subspaces is greater that 1. We show that every Banach space contains a Diestel subspace and that minimal Banach spaces are Diestel spaces.
</p>projecteuclid.org/euclid.bjma/1487732418_20170221220035Tue, 21 Feb 2017 22:00 ESTRadon–Nikodym theorems for operator-valued measures and continuous generalized frameshttp://projecteuclid.org/euclid.bjma/1487732419<strong>Fengjie Li</strong>, <strong>Pengtong Li</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 2, 363--381.</p><p><strong>Abstract:</strong><br/>
In this article we determine that an operator-valued measure (OVM) for Banach spaces is actually a weak∗ measure, and then we show that an OVM can be represented as an operator-valued function if and only if it has $\sigma$ -finite variation. By the means of direct integrals of Hilbert spaces, we introduce and investigate continuous generalized frames ( continuous operator-valued frames , or simply CG frames ) for general Hilbert spaces. It is shown that there exists an intrinsic connection between CG frames and positive OVMs. As a byproduct, we show that a Riesz-type CG frame does not exist unless the associated measure space is purely atomic. Also, a dilation theorem for dual pairs of CG frames is given.
</p>projecteuclid.org/euclid.bjma/1487732419_20170221220035Tue, 21 Feb 2017 22:00 ESTOn a generalized Šemrl’s theorem for weak 2-local derivations on $B(H)$http://projecteuclid.org/euclid.bjma/1488877211<strong>Juan Carlos Cabello</strong>, <strong>Antonio M. Peralta</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 2, 382--397.</p><p><strong>Abstract:</strong><br/>
We prove that, for every complex Hilbert space $H$ , every weak 2-local derivation on $B(H)$ or on $K(H)$ is a linear derivation. We also establish that every weak 2-local derivation on an atomic von Neumann algebra or on a compact C $^{*}$ -algebra is a linear derivation.
</p>projecteuclid.org/euclid.bjma/1488877211_20170307040039Tue, 07 Mar 2017 04:00 ESTLattice properties of the core-partial orderhttp://projecteuclid.org/euclid.bjma/1488877212<strong>Marko S. Djikić</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 2, 398--415.</p><p><strong>Abstract:</strong><br/>
We show that in an arbitrary Hilbert space, the set of group-invertible operators with respect to the core-partial order has the complete lower semilattice structure, meaning that an arbitrary family of operators possesses the core-infimum. We also give a necessary and sufficient condition for the existence of the core-supremum of an arbitrary family, and we study the properties of these lattice operations on pairs of operators.
</p>projecteuclid.org/euclid.bjma/1488877212_20170307040039Tue, 07 Mar 2017 04:00 ESTAdditive maps preserving Drazin invertible operators of index $n$http://projecteuclid.org/euclid.bjma/1489802494<strong>Mostafa Mbekhta</strong>, <strong>Mourad Oudghiri</strong>, <strong>Khalid Souilah</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 2, 416--437.</p><p><strong>Abstract:</strong><br/>
Given an integer $n\geq2$ , in this article we provide a complete description of all additive surjective maps on the algebra of all bounded linear operators acting on an infinite-dimensional complex Banach space, preserving in both directions the set of Drazin invertible operators of index $n$ .
</p>projecteuclid.org/euclid.bjma/1489802494_20170317220159Fri, 17 Mar 2017 22:01 EDTOn some Hardy-type inequalities for fractional calculus operatorshttp://projecteuclid.org/euclid.bjma/1489802495<strong>Sajid Iqbal</strong>, <strong>Josip Pečarić</strong>, <strong>Muhammad Samraiz</strong>, <strong>Zivorad Tomovski</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 2, 438--457.</p><p><strong>Abstract:</strong><br/>
In this article we present applications of Hardy-type and refined Hardy-type inequalities for a generalized fractional integral operator involving the Mittag-Leffler function in its kernel and for the Hilfer fractional derivative using convex and monotone convex functions.
</p>projecteuclid.org/euclid.bjma/1489802495_20170317220159Fri, 17 Mar 2017 22:01 EDTDisjoint hypercyclic weighted translations on groupshttp://projecteuclid.org/euclid.bjma/1492618124<strong>Chung-Chuan Chen</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 3, 459--476.</p><p><strong>Abstract:</strong><br/>
Let $1\leq p\lt \infty$ , and let $G$ be a locally compact group. We characterize disjoint hypercyclic weighted translation operators on the Lebesgue space $L^{p}(G)$ in terms of the weight, the Haar measure, and the group element. Disjoint supercyclic, disjoint mixing, and dual disjoint hypercyclic weighted translation operators are also characterized.
</p>projecteuclid.org/euclid.bjma/1492618124_20170718220400Tue, 18 Jul 2017 22:04 EDTHardy-type space estimates for multilinear commutators of Calderón–Zygmund operators on nonhomogeneous metric measure spaceshttp://projecteuclid.org/euclid.bjma/1492618125<strong>Jie Chen</strong>, <strong>Haibo Lin</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 3, 477--496.</p><p><strong>Abstract:</strong><br/>
Let $(\mathcal{X},d,\mu)$ be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition . Let $T$ be a Calderón–Zygmund operator and let $\vec{b}:=(b_{1},\ldots,b_{m})$ be a finite family of $\operatorname{\widetilde{RBMO}}(\mu)$ functions. In this article, the authors establish the boundedness of the multilinear commutator $T_{\vec{b}}$ , generated by $T$ and $\vec{b}$ from the atomic Hardy-type space $\widetilde{H}_{\mathrm{fin},\vec{b},m,\rho}^{1,q,m+1}(\mu)$ into the Lebesgue space $L^{1}(\mu)$ . The authors also prove that $T_{\vec{b}}$ is bounded from the atomic Hardy-type space $\widetilde{H}_{\mathrm{fin},\vec{b},m,\rho}^{1,q,m+2}(\mu)$ into the atomic Hardy space $\widetilde{H}^{1}(\mu)$ via the molecular characterization of $\widetilde{H}^{1}(\mu)$ .
</p>projecteuclid.org/euclid.bjma/1492618125_20170718220400Tue, 18 Jul 2017 22:04 EDTOn the existence of at least a solution for functional integral equations via measure of noncompactnesshttp://projecteuclid.org/euclid.bjma/1492618126<strong>Calogero Vetro</strong>, <strong>Francesca Vetro</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 3, 497--512.</p><p><strong>Abstract:</strong><br/>
In this article, we use fixed-point methods and measure of noncompactness theory to focus on the problem of establishing the existence of at least a solution for the following functional integral equation
\[u(t)=g(t,u(t))+\int_{0}^{t}G(t,s,u(s))\,ds,\quad t\in{[0,+\infty[},\] in the space of all bounded and continuous real functions on $\mathbb{R}_{+}$ , under suitable assumptions on $g$ and $G$ . Also, we establish an extension of Darbo’s fixed-point theorem and discuss some consequences.
</p>projecteuclid.org/euclid.bjma/1492618126_20170718220400Tue, 18 Jul 2017 22:04 EDTWeighted Herz space estimates for Hausdorff operators on the Heisenberg grouphttp://projecteuclid.org/euclid.bjma/1492618127<strong>Jianmiao Ruan</strong>, <strong>Dashan Fan</strong>, <strong>Qingyan Wu</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 3, 513--535.</p><p><strong>Abstract:</strong><br/>
In this article, we study the Hausdorff operator, defined via a general linear mapping $A$ , on weighted Herz spaces in the setting of the Heisenberg group. Under some assumptions on the mapping $A$ , we establish its sharp boundedness on power-weighted Herz spaces and power-weighted Lebesgue spaces in the Heisenberg group. Our proof is heavily based on the block decomposition of the Herz space, which is quite different from any other function spaces. Our results extend and improve some existing theorems.
</p>projecteuclid.org/euclid.bjma/1492618127_20170718220400Tue, 18 Jul 2017 22:04 EDTToeplitz algebras arising from escape points of interval mapshttp://projecteuclid.org/euclid.bjma/1493431219<strong>C. Correia Ramos</strong>, <strong>Nuno Martins</strong>, <strong>Paulo R. Pinto</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 3, 536--553.</p><p><strong>Abstract:</strong><br/>
We generate a representation of the Toeplitz $C^{\ast}$ -algebra $\mathcal{T}_{A_{f}}$ on a Hilbert space $H_{x}$ that encodes the orbit of an escape point $x\in I$ of a Markov interval map $f$ , with transition matrix $A_{f}$ . This leads to a family of representations of $\mathcal{T}_{A_{f}}$ labeled by points in all intervals $I$ . The underlying dynamics of the interval map are used in the study of this family.
</p>projecteuclid.org/euclid.bjma/1493431219_20170718220400Tue, 18 Jul 2017 22:04 EDTTwo-sided and one-sided invertibility of Wiener-type functional operators with a shift and slowly oscillating datahttp://projecteuclid.org/euclid.bjma/1493776976<strong>Gustavo Fernández-Torres</strong>, <strong>Yuri Karlovich</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 3, 554--590.</p><p><strong>Abstract:</strong><br/>
Let $\alpha$ be an orientation-preserving homeomorphism of $[0,\infty]$ onto itself with only two fixed points at $0$ and $\infty$ , whose restriction to $\mathbb{R}_{+}=(0,\infty)$ is a diffeomorphism, and let $U_{\alpha}$ be the isometric shift operator acting on the Lebesgue space $L^{p}(\mathbb{R}_{+})$ with $p\in[1,\infty]$ by the rule $U_{\alpha}f=(\alpha')^{1/p}(f\circ\alpha)$ . We establish criteria of the two-sided and one-sided invertibility of functional operators of the form \begin{equation}A=\sum_{k\in\mathbb{Z}}a_{k}U_{\alpha}^{k}\quadwhere\Vert A\Vert _{W}=\sum_{k\in\mathbb{Z}}\Vert a_{k}\Vert _{L^{\infty}(\mathbb{R}_{+})}\lt \infty,\end{equation} on the spaces $L^{p}(\mathbb{R}_{+})$ under the assumptions that the functions $\log\alpha'$ and $a_{k}$ for all $k\in\mathbb{Z}$ are bounded and continuous on $\mathbb{R}_{+}$ and may have slowly oscillating discontinuities at $0$ and $\infty$ . The unital Banach algebra $\mathfrak{A}_{W}$ of such operators is inverse-closed: if $A\in\mathfrak{A}_{W}$ is invertible on $L^{p}(\mathbb{R}_{+})$ for $p\in[1,\infty]$ , then $A^{-1}\in\mathfrak{A}_{W}$ . Obtained criteria are of two types: in terms of the two-sided or one-sided invertibility of so-called discrete operators on the spaces $l^{p}$ and in terms of conditions related to the fixed points of $\alpha$ and the orbits $\{\alpha^{n}(t):n\in\mathbb{Z}\}$ of points $t\in\mathbb{R}_{+}$ .
</p>projecteuclid.org/euclid.bjma/1493776976_20170718220400Tue, 18 Jul 2017 22:04 EDTA generalization of Kantorovich operators for convex compact subsetshttp://projecteuclid.org/euclid.bjma/1494036023<strong>Francesco Altomare</strong>, <strong>Mirella Cappelletti Montano</strong>, <strong>Vita Leonessa</strong>, <strong>Ioan Raşa</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 3, 591--614.</p><p><strong>Abstract:</strong><br/>
In this article, we introduce and study a new sequence of positive linear operators acting on function spaces defined on a convex compact subset. Their construction depends on a given Markov operator, a positive real number, and a sequence of Borel probability measures. By considering special cases of these parameters for particular convex compact subsets, we obtain the classical Kantorovich operators defined in the $1$ -dimensional and multidimensional setting together with several of their wide-ranging generalizations scattered in the literature. We investigate the approximation properties of these operators by also providing several estimates of the rate of convergence. Finally, we discuss the preservation of Lipschitz-continuity and of convexity.
</p>projecteuclid.org/euclid.bjma/1494036023_20170718220400Tue, 18 Jul 2017 22:04 EDTThe multiplier algebra of the noncommutative Schwartz spacehttp://projecteuclid.org/euclid.bjma/1494036021<strong>Tomasz Ciaś</strong>, <strong>Krzysztof Piszczek</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 3, 615--635.</p><p><strong>Abstract:</strong><br/>
We describe the multiplier algebra of the noncommutative Schwartz space. This multiplier algebra can be seen as the largest $^{*}$ -algebra of unbounded operators on a separable Hilbert space with the classical Schwartz space of rapidly decreasing functions as the domain. We show in particular that it is neither a $\mathcal{Q}$ -algebra nor $m$ -convex. On the other hand, we prove that classical tools of functional analysis, for example, the closed graph theorem, the open mapping theorem, or the uniform boundedness principle, are still available.
</p>projecteuclid.org/euclid.bjma/1494036021_20170718220400Tue, 18 Jul 2017 22:04 EDTNormed Orlicz function spaces which can be quasi-renormed with easily calculable quasinormshttp://projecteuclid.org/euclid.bjma/1496973700<strong>Paweł Foralewski</strong>, <strong>Henryk Hudzik</strong>, <strong>Radosław Kaczmarek</strong>, <strong>Miroslav Krbec</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 3, 636--660.</p><p><strong>Abstract:</strong><br/>
We are interested in the widest possible class of Orlicz functions $\Phi$ such that the easily calculable quasinorm $[f]_{\Phi,p}:=\Vert f\Vert_{E}\{I_{\Phi}(\frac{f}{\Vertf\Vert_{E}})\}^{1\slash p}$ if $f\neq0$ and $[f]_{\Phi,p}=0$ if $f=0$ , on the Orlicz space $L^{\Phi}(\Omega,\Sigma,\mu)$ generated by $\Phi$ , is equivalent to the Luxemburg norm $\Vert\cdot\Vert_{\Phi}$ . To do this, we use a suitable $\Delta_{2}$ -condition, lower and upper Simonenko indices $p_{S}^{a}(\Phi)$ and $q_{S}^{a}(\Phi)$ for the generating function $\Phi$ , numbers $p\in[1,p_{S}^{a}(\Phi)]$ satisfying $q_{S}^{a}(\Phi)-p\leq1$ , and an embedding of $L^{\Phi}(\Omega,\Sigma,\mu)$ into a suitable Köthe function space $E=E(\Omega,\Sigma,\mu)$ . We take as $E$ the Lebesgue spaces $L^{r}(\Omega,\Sigma,\mu)$ with $r\in[1,p_{S}^{l}(\Phi)]$ , when the measure $\mu$ is nonatomic and finite, and the weighted Lebesgue spaces $L^{r}_{\omega}(\Omega,\Sigma,\mu)$ , with $r\in[1,p_{S}^{a}(\Phi)]$ and a suitable weight function $\omega$ , when the measure $\mu$ is nonatomic infinite but $\sigma$ -finite. We also use condition $\nabla_{3}$ if $p_{S}^{a}(\Phi)=1$ and condition $\nabla^{2}$ if $p_{S}^{a}(\Phi)\gt 1$ , proving their necessity in most of the considered cases. Our results seem important for applications of Orlicz function spaces.
</p>projecteuclid.org/euclid.bjma/1496973700_20170718220400Tue, 18 Jul 2017 22:04 EDTA hybrid shrinking projection method for common fixed points of a finite family of demicontractive mappings with variational inequality problemshttp://projecteuclid.org/euclid.bjma/1495505201<strong>Suthep Suantai</strong>, <strong>Withun Phuengrattana</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 3, 661--675.</p><p><strong>Abstract:</strong><br/>
In this article, we prove some properties of a demicontractive mapping defined on a nonempty closed convex subset of a Hilbert space. By using these properties, we obtain strong convergence theorems of a hybrid shrinking projection method for finding a common element of the set of common fixed points of a finite family of demicontractive mappings and the set of common solutions of a finite family of variational inequality problems in a Hilbert space. A numerical example is presented to illustrate the proposed method and convergence result. Our results improve and extend the corresponding results existing in the literature.
</p>projecteuclid.org/euclid.bjma/1495505201_20170718220400Tue, 18 Jul 2017 22:04 EDTTernary weak amenability of the bidual of a JB $^{*}$ -triplehttp://projecteuclid.org/euclid.bjma/1496973699<strong>Mohsen Niazi</strong>, <strong>Mohammad Reza Miri</strong>, <strong>Hamid Reza Ebrahimi Vishki</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 3, 676--697.</p><p><strong>Abstract:</strong><br/>
Beside the triple product induced by ultrapowers on the bidual of a JB $^{*}$ -triple, we assign a triple product to the bidual, $E^{**}$ , of a JB-triple system $E$ , and we show that, under some mild conditions, it makes $E^{**}$ a JB-triple system. To study ternary $n$ -weak amenability of $E^{**}$ , we need to improve the module structures in the category of JB-triple systems and their iterated duals, which lead us to introduce a new type of ternary module. We then focus on the main question: when does ternary $n$ -weak amenability of $E^{**}$ imply the same property for $E?$ In this respect, we show that if the bidual of a JB $^{*}$ -triple $E$ is ternary $n$ -weakly amenable, then $E$ is ternary $n$ -quasiweakly amenable. However, for a general JB-triple system, the results are slightly different for $n=1$ and $n\geq2$ , and the case $n=1$ requires some additional assumptions.
</p>projecteuclid.org/euclid.bjma/1496973699_20170718220400Tue, 18 Jul 2017 22:04 EDTOn the universal and strong $(L^{1},L^{\infty})$ -property related to Fourier–Walsh serieshttp://projecteuclid.org/euclid.bjma/1494900292<strong>Martin G. Grigoryan</strong>. <p><strong>Source: </strong>Banach Journal of Mathematical Analysis, Volume 11, Number 3, 698--712.</p><p><strong>Abstract:</strong><br/>
In this article, we construct a function $U\in L^{1}[0,1)$ with strictly decreasing Fourier–Walsh coefficients $\{c_{k}(U)\}\searrow$ , and having a universal and strong $(L^{1},L^{\infty})$ -property with respect to the Walsh system.
</p>projecteuclid.org/euclid.bjma/1494900292_20170718220400Tue, 18 Jul 2017 22:04 EDT