Showing papers in "Banach Journal of Mathematical Analysis in 2015"
TL;DR: A survey and a discussion of selected recent results concerning such stability of the equations of homomorphisms, focussing especially on some conditional versions of them is given in this article.
Abstract: The issue of Ulam's type stability of an equation is understood in the following way: when a mapping which satisfies the equation approximately (in some sense), it is "close" to a solution of it. In this expository paper, we present a survey and a discussion of selected recent results concerning such stability of the equations of homomorphisms, focussing especially on some conditional versions of them.
61 citations
TL;DR: In this paper, the authors first introduced the Triebel-Lizorkin-type space with variable exponents, and established its transformation characterization in the sense of Frazier and Jawerth.
Abstract: In this article, the authors first introduce the Triebel--Lizorkin-type space $F_{p(\cdot),q(\cdot)}^{s(\cdot),\phi}(\mathbb R^n)$ with variable exponents, and establish its $\varphi$-transform characterization in the sense of Frazier and Jawerth, which further implies that this new scale of function spaces is well defined. The smooth molecular and the smooth atomic characterizations of $F_{p(\cdot),q(\cdot)}^{s(\cdot),\phi}(\mathbb R^n)$ are also obtained, which are used to prove a trace theorem of $F_{p(\cdot),q(\cdot)}^{s(\cdot),\phi}(\mathbb R^n)$. The authors also characterize the space $F_{p(\cdot),q(\cdot)}^{s(\cdot),\phi}(\mathbb R^n)$ via Peetre maximal functions.
49 citations
TL;DR: In this article, some operator inequalities for positive linear maps are presented, which improve and generalize the corresponding results due to Fu and He [Linear Multilinear Algebra, doi: 10.1080/03081087.2014.880432].
Abstract: Some operator inequalities for positive linear maps are presented. These inequalities improve and generalize the corresponding results due to Fu and He [Linear Multilinear Algebra, doi: 10.1080/03081087.2014.880432.].
21 citations
TL;DR: In this paper, the basic tools of frame theory are described in the formalism of Krein spaces, and it is shown how to transfer a frame for Hilbert spaces to Krein space given by a $W$-metric, where the Gram operator is not necessarily regular and possibly unbounded.
Abstract: A definition of frames in Krein spaces is stated and a complete characterization is given by comparing them to frames in the associated Hilbert space. The basic tools of frame theory are described in the formalism of Krein spaces. It is shown how to transfer a frame for Hilbert spaces to Krein spaces given by a $W$-metric, where the Gram operator $W$ is not necessarily regular and possibly unbounded.
19 citations
TL;DR: Zametki et al. as discussed by the authors showed that it is possible to solve homogenization problems in nonergodic algebras with mean value regardless of whether they are ergodic or not.
Abstract: In several works, the theory of strongly continuous groups is used to build a framework for solving stochastic homogenization problems. Following this idea, we construct a detailed and comprehensive theory of homogenization. This enables to solve homogenization problems in algebras with mean value, regardless of whether they are ergodic or not, thereby responding affirmatively to the question raised by Zhikov and Krivenko [V.V. Zhikov, E.V. Krivenko, Homogenization of singularly perturbed elliptic operators. Matem. Zametki, 33 (1983) 571-582 (english transl.: Math. Notes, 33 (1983) 294-300)] to know whether it is possible to homogenize problems in nonergodic algebras. We also state and prove a compactness result for Young measures in these algebras. As an important achievement we study the homogenization problem associated with a stochastic Ladyzhenskaya model for incompressible viscous flow, and we present and solve a few examples of homogenization problems related to nonergodic algebras.
19 citations
TL;DR: In this article, the boundedness of the intrinsic square functions and their commutators on generalized weighted Orlicz-Morrey spaces was investigated in terms of Zygmund-type integral inequalities on weights.
Abstract: We shall investigate the boundedness of the intrinsic square functions and their commutators on generalized weighted Orlicz--Morrey spaces $M^{\Phi,\varphi}_{w}({\mathbb{R}}^n)$. In all the cases, the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on weights $\varphi$ without assuming any monotonicity property of $\varphi(x,\cdot)$ with $x$ fixed.
19 citations
TL;DR: In this paper, the boundedness of vector-valued sublinear operators in Herz-Morrey spaces with variable exponents was studied and the equivalent quasi-norms on these spaces were proved by Peetre's maximal operators.
Abstract: In the article, the boundedness of vector-valued sublinear operators in Herz--Morrey spaces with variable exponents $M\dot{K}^{\alpha(\cdot),\lambda}_{q,p(\cdot)}(\mathbb{R}^{n})$ are obtained. Then Herz--Morrey type Besov and Triebel-Lizorkin spaces with variable exponents are introduced. Finally, we prove the equivalent quasi-norms on these spaces by Peetre's maximal operators.
18 citations
18 citations
TL;DR: In this paper, the spectral radius inequalities for nonnegative matrices using the ideas of Audenaert and the Cauchy-Schwarz inequality for spectral radius were presented.
Abstract: We present some spectral radius inequalities for nonnegative matrices Using the ideas of Audenaert, we then prove the inequality which may be regarded as a Cauchy--Schwarz inequality for spectral radius of nonnegative matrices $$ \rho(A \circ B) \leq [\rho(A \circ A)]^{\frac{1}{2}}[\rho(B\circ B)]^{\frac{1}{2}} $$ In addition, new proofs of some related results due to Horn and Zhang, Huang are also given Finally, we interpolate Huang's inequality by proving $$ \rho(A_{1}\circ A_{2} \circ \cdots \circ A_{k}) \leq [\rho(A_{1}A_2\cdots A_{k})]^{1-\frac{2}{k}}[\rho((A_{1}\circ A_{1})\cdots (A_{k}\circ A_{k})]^{\frac{1}{k}} \leq \rho(A_{1}A_2 \cdots A_{k})$$
15 citations
TL;DR: In this article, the authors obtained some coincidence and periodic points results for generalized $F$-type contractions in a metric space endowed with a graph, and applied their results to establishing the existence of solution for a certain type of nonlinear integral equation.
Abstract: The purpose of this paper is to obtain some coincidence and periodic points results for generalized $F$-type contractions in a metric space endowed with a graph. Some examples are given to illustrate the new theory. Then, we apply our results to establishing the existence of solution for a certain type of nonlinear integral equation.
14 citations
TL;DR: In this article, the authors introduce Bessel multipliers, g-multipliers, and Bessel fusion multipliers in Hilbert modules and show that they share many useful properties with their corresponding notions in Hilbert and Banach spaces.
Abstract: In this paper we introduce Bessel multipliers, g-Bessel multipliers and Bessel fusion multipliers in Hilbert $C^\ast$--modules and we show that they share many useful properties with their corresponding notions in Hilbert and Banach spaces. We show that various properties of multipliers are closely related to their symbols and Bessel sequences, especially we consider multipliers when their Bessel sequences are modular Riesz bases and we see that in this case multipliers can be composed and inverted. We also study bounded below multipliers and generalize some of the results obtained for fusion frames in Hilbert spaces to Hilbert $C^\ast$--modules.
TL;DR: In this article, the sharp order is extended to the set of elements for which left and right annihilators are respectively principal left and principal right ideals generated by the same idempotent.
Abstract: The sharp order is a well known partial order defined on the set of complex matrices with index less or equal one. Following Semrl's approach, Efimov extended this order to the set of those bounded Banach space operators $A$ for which the closure of the image and kernel are topologically complementary subspaces. In order to extend the sharp order to arbitrary ring $R$ (particulary to Rickart and Rickart $*$-rings) we use the notions of annihilators. The concept of the sharp order is extended to the set $\mathcal{I}_R$ of those elements for which left and right annihilators are respectively principal left and principal right ideals generated by the same idempotent. It is proved that the sharp order is a partial order relation on $\mathcal{I}_R$. Following the idea we also extend and discuss the recently introduced concept of core partial order.
TL;DR: The complete characterization of Hardy-type inequalities on the weighted cones of quasi-concave functions for all $p,q \in (0,\infty)$ is given in this article.
Abstract: The complete characterization of the Hardy-type $L^p - L^q$ inequalities on the weighted cones of quasi-concave functions for all $p,q \in (0,\infty)$ is given.
TL;DR: In this article, the Hermite-Hadamard type inequality for operator preinvex functions of self-adjoint operators in Hilbert spaces was established and an estimate of the right hand side of a Hermite Hadamard Type Inequality was given.
Abstract: In this paper we introduce operator preinvex functions and establish a Hermite--Hadamard type inequality for such functions. We give an estimate of the right hand side of a Hermite--Hadamard type inequality in which some operator preinvex functions of selfadjoint operators in Hilbert spaces are involved. Also some Hermite--Hadamard type inequalities for the product of two operator preinvex functions are given.
TL;DR: The existence of a linear isometric correspondence between the Banach space of all symmetric orthogonal forms on a JB$^*$-algebra and the corresponding Banach spaces of all purely Jordan generalized Jordan derivations from $\mathcal{J}$ into √ √ {J} was shown in this paper.
Abstract: We prove the existence of a linear isometric correspondence between the Banach space of all symmetric orthogonal forms on a JB$^*$-algebra $\mathcal{J}$ and the Banach space of all purely Jordan generalized Jordan derivations from $\mathcal{J}$ into $\mathcal{J}^*$. We also establish the existence of a similar linear isometric correspondence between the Banach spaces of all anti-symmetric orthogonal forms on $\mathcal{J}$, and of all Lie Jordan derivations from $\mathcal{J}$ into $\mathcal{J}^*$.
TL;DR: The main purpose of as mentioned in this paper is to develop the generalized analytic Fourier-Feynman transform theory, and to establish a relationship between these two generalized analytic transforms, which can be seen as a generalization of Fourier transform theory.
Abstract: The main purpose of this article is to develop the generalized analytic Fourier--Feynman transform theory. We introduce a generalized analytic Fourier--Feynman transform and a multiple generalized analytic Fourier--Feynman transform with respect to Gaussian processes on the function space $C_{a,b}[0,T]$ induced by a generalized Brownian motion process. We then establish a relationship between these two generalized analytic transforms.
TL;DR: In this article, a canonical decomposition for general skew symmetric weighted shifts is given, and based on this decomposition, a classification of skew-symmetric weighted shift operators is provided.
Abstract: An operator $T$ on a complex Hilbert space $\mathcal{H}$ is called skew symmetric if $T$ can be represented as a skew symmetric matrix relative to some orthonormal basis for $\mathcal{H}$. We first give a canonical decomposition for general skew symmetric operators. Based on this decomposition, we provide a classification of skew symmetric weighted shifts.
TL;DR: In this article, weighted composition operators between weak and strong vector-valued Bergman spaces and Hardy spaces are investigated, and some estimates of their norms are given; see Section 2.1.
Abstract: In this paper we investigate weighted composition operators between weak and strong vector-valued Bergman spaces and Hardy spaces, and give some estimates of their norms
TL;DR: In this article, a few matrix splittings arising from index-proper splitting are introduced and their convergence results and their applications to multisplitting theory are studied.
Abstract: Matrix splitting and its convergence theorems are useful tools for finding solution of linear system of equations, iteratively. In this article, we introduce a few matrix splittings arising from index-proper splittings. Then their convergence results and their applications to multisplitting theory are studied.
TL;DR: In this paper, it was shown that the twisted crossed product is also a symmetric Banach-algebra, for any twisted action of the projective tensor product of a discrete group.
Abstract: A discrete group $\mathrm{G}$ is called rigidly symmetric if for every $C^*$-algebra $\mathcal{A}$ the projective tensor product $\ell^1(\mathrm{G})\widehat\otimes A$ is a symmetric Banach $^*$-algebra. For such a group we show that the twisted crossed product $\ell^1_{\alpha,\omega}(\mathrm{G};\mathcal{A})$ is also a symmetric Banach $^*$-algebra, for every twisted action $(\alpha,\omega)$ of $\mathrm{G}$ in a $C^*$-algebra $\mathcal{A}$. We extend this property to other types of decay, replacing the $\ell^1$-condition. We also make the connection with certain classes of twisted kernels, used in a theory of integral operators involving group $2$-cocycles. The algebra of these kernels is studied, both in intrinsic and in represented version.
TL;DR: In this article, the Lipschitz Grothendieck integral operators which are exactly those elements in the Lip-free space were studied, and they correspond to the elements of the dual of the injective tensor product of the lipschitzer-free spaces.
Abstract: Let $X$ be a pointed metric space and let $E$ be a Banach space. It is known that the Lipschitz space $\mathrm{Lip}_o(X,E^*)$ is isometrically isomorphic to $(\mathcal{F}(X)\widehat{\otimes}_\pi E)^*$, the dual of the projective tensor product of the Lipschitz-free space $\mathcal{F}(X)$ and $E$. Since the injective norm $\varepsilon$ on $\mathcal{F}(X)\otimes E$ is smaller than the projective norm $\pi$, we study Lipschitz Grothendieck-integral operators which are exactly those elements in $\mathrm{Lip}_o(X,E^*)$ which correspond to the elements of $(\mathcal{F}(X)\widehat{\otimes}_\varepsilon E)^*$, the dual of the injective tensor product of $\mathcal{F}(X)$ and $E$.
TL;DR: In this paper, a sufficient criterion for a continuous operator to be frequently hypercyclic on a locally convex space is derived and applications are given to obtain $q$-frequently hyper cyclic operators with respect to the norm-, $F$-norm- and weak*- topologies.
Abstract: We introduce $q$-frequently hypercyclic operators and derive a sufficient criterion for a continuous operator to be $q$-frequently hypercyclic on a locally convex space. Applications are given to obtain $q$-frequently hypercyclic operators with respect to the norm-, $F$-norm- and weak*- topologies. Finally, the frequent hypercyclicity of the non-convolution operator $T_\mu$ defined by $T_\mu(f)(z)=f'(\mu z)$, $|\mu|\geq 1$ on the space $H(\mathbb{C})$ of entire functions equipped with the compact-open topology is shown.
TL;DR: In this paper, the authors studied the noncommutative Orlicz modular spaces associated with growth functions and proved Young inequalities and Clarkson-McCarthy inequalities on these spaces.
Abstract: We study the noncommutative Orlicz modular spaces associated with growth functions. Some basic properties of such spaces, such as completeness and dominated convergence theorem, are present. Moreover, Young inequalities and Clarkson--McCarthy inequalities on these spaces proved.
TL;DR: In this article, the concepts of open unit ball and closed unit ball in a real or complex normed space are naturally extended to the scope of topological rings with unity, and a type of open (closed) sets called open closed unit neighborhoods of 0 are defined.
Abstract: The concepts of open unit ball and closed unit ball in a real or complex normed space are naturally extended to the scope of topological rings with unity. We then define a type of open (closed) sets called open (closed) unit neighborhoods of 0. We show among other things that in R and C the only non-trivial open and closed unit neighborhoods of 0 are the open unit ball and the closed unit ball, respectively.
TL;DR: In this article, the authors introduce an exact and approximate orthogonality relation between linear mappings and linear orthogonalities, and show that every linear mapping that approximately preserves the $B$-orthogonality is also a scalar multiple of an almost isometry.
Abstract: In a normed space we introduce an exact and approximate orthogonality relation. We consider classes of linear mappings approximately preserving this kind of orthogonality. We show that, in particular, the property that a linear mapping approximately preserves the $B$-orthogonality is equivalent to that it approximately preserves the $\rho,\rho_+$-orthogonality (although these orthogonalities need not be equivalent). Moreover, we show that every approximately orthogonality preserving linear mapping is necessarily a scalar multiple of an almost isometry.
TL;DR: The work in this paper was partially supported by the Fundacao para a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) through the project PEst-OE/MAT/UIO297/2014 (Centro de Matemdtica e Aplicacoes).
Abstract: This work was partially supported by the Fundacao para a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) through the project PEst-OE/MAT/UIO297/2014 (Centro de Matemdtica e Aplicacoes). The authors would like to thank the anonymous referee for useful remarks and for informing about the work [5].
TL;DR: In this article, the authors present characterizations for absolute continuity of representable positive functionals on general $^*$-algebras and give a new and very different proof to the Lebesgue decomposition theorem for representable functions.
Abstract: The goal of this paper is to present characterizations for absolute continuity of representable positive functionals on general $^*$-algebras. From the results we give a new and very different proof to our recently published Lebesgue decomposition theorem for representable positive functionals. On unital $C^*$-algebras and measure algebras of compact groups further characterizations are included in the paper. As an application of our results, we answer Gudder's problem on the uniqueness of the Lebesgue decomposition in the case of commutative $^*$-algebras and measure algebras of compact groups. Another application to faithful positive functionals defined on the latter $^*$-algebras is also included.
TL;DR: In this article, the authors studied disjoint, strongly and weakly disjoointness of $g$-frames in a Hilbert space and provided necessary and sufficient conditions for disjoining.
Abstract: In this paper, we study disjoint, strongly disjoint and weakly disjoint $g$-frames in Hilbert spaces and we provide necessary and sufficient conditions for disjointness, strongly disjointness and weakly disjointness of $g$-frames. Also, by using the orthogonal projections in Hilbert spaces, we prove that dual $g$-frames for a Hilbert space can be dilated to a $g$-Riesz basis for some larger Hilbert space and its dual $g$-Riesz basis.
TL;DR: In this paper, the authors present a sufficient condition for a Banach space to have the approximate hyperplane series property (AHSP) which actually covers all known examples and use this property to get a stability result to vector-valued spaces of integrable functions.
Abstract: We present a sufficient condition for a Banach space to have the approximate hyperplane series property (AHSP) which actually covers all known examples. We use this property to get a stability result to vector-valued spaces of integrable functions. On the other hand, the study of a possible Bishop--Phelps--Bollobas version of a classical result of V. Zizler leads to a new characterization of the AHSP for dual spaces in terms of $w^*$-continuous operators and other related results.