Some more weak Hilbert spaces

Structure of octonionic Hilbert spaces with applications in the Parseval equality and Cayley-Dickson algebras

Contrary to the simple structure of the tensor product of the quaternionic Hilbert space, the octonionic situation becomes more involved. It turns out that an octonionic Hilbert space can be decomposed as an orthogonal direct sum of two subspaces, each of them isomorphic to a tensor product of an irreducible octonionic Hilbert space with a real Hilbert space. As an application, we find that for a given orthogonal basis the octonionic Parseval equality holds if and only if the basis is weak associative. Fortunately, there always exists a weak associative orthogonal basis in an octonionic Hilber…

On the ergodic theorem for affine actions on Hilbert space

This note establishes a new weak mean ergodic theorem for 1-cocycles associated to weakly mixing representations of amenable groups.

The number of weakly compact convex subsets of the Hilbert space

We prove that for k an uncountable cardinal, there exist 2^k many non homeomorphic weakly compact convex subsets of weight k in the Hilbert space of density k.

An example of weakly amenable and character amenable operator

A complete characterization of Hilbert space operators that generate weakly amenable algebras remains open, even in the case of compact operator. Farenick, Forrest and Marcoux proposed the question that if $T$ is a compact weakly amenable operator on a Hilbert space $\mathfrak{H}$, then is $T$ similar to a normal operator? In this paper we demonstrate an example of compact triangular operator on infinite-dimension Hilbert space which is a weakly amenable and character amenable operator but is not similar to a normal operator.

A note on (weak) phase and norm retrievable Real Hilbert space frames and projections

\begin{abstract} In this manuscript, we answer a list of longstanding open problems on weak phase retrieval including: (1) A complete classification of the vectors $\{x_i\}_{i=1}^2$ in $\RR^3$ that do weak phase retrieval; (2) We show that frames doing weak phase retrieval in $\RR^n$ must span $\RR^n$; (3) We give an example of a set of vectors doing phase retrieval but their orthogonal complement hyperplanes fail weak phase retrieval; (4) We give a classification of weak phase retrievable frames - which makes clear the difference between phase retrieval and weak phase retrieval; (5) We classi…

Module weak Banach-Saks and module Schur properties of Hilbert C*-modules

Continuing the research on the Banach-Saks and Schur properties started in (cf. M. Frank, A. A. Pavlov, Banach-Saks properties of C*-algebras and Hilbert C*-modules (submitted)) we investigate analogous properties in the module context. As an environment serves the class of Hilbert C*-modules. Some properties of weak module topologies on Hilbert C*-modules are described. Natural module analogues of the classical weak Banach-Saks and the classical Schur properties are defined and studied. A number of useful characterizations of properties of Hilbert C*-modules is obtained. In particular, some i…

The unit ball of the Hilbert space in its weak topology

We show that the unit ball of a Hilbert space in its weak topology is a continuous image of the countable power of the Alexandroff compactification of a discrete set, and we deduce some combinatorial properties of its lattice of open sets which are not shared by the balls of other equivalent norms when the space is nonseparable.

Weak and strong approximation of semigroups on Hilbert spaces

For a sequence of uniformly bounded, degenerate semigroups on a Hilbert space, we compare various types of convergences to a limit semigroup. Among others, we show that convergence of the semigroups, or of the resolvents of the generators, in the weak operator topology, in the strong operator topology or in certain integral norms are equivalent under certain natural assumptions which are frequently met in applications.

关于算子弱稳定性的阐述

An Exposition on Weak Stability of Operators

本文综述赋范空间（尤其是希尔伯特空间）上有界线性算子的弱稳定性研究，梳理六十年文献并指出研究集中于希尔伯特空间算子及酉算子弱稳定性的核心地位。

Some Levin-Steckin's Type Inequalities for Operator Convex Functions on Hilbert Spaces

In this paper we obtain some operator versions of Levin-Steckin integral inequality.

Refinements of Some Numerical radius inequalities for Hilbert Space Operators

In this work, some generalizations and refinements inequalities for numerical radius of the product of Hilbert space operators are proved. New inequalities for numerical radius of block matrices of Hilbert space operators are also established.

Some properties of K-frame in n-Hilbert space

The notion of a K-frame in n-Hilbert space is presented and some of their characterizations are given. We verify that sum of two K-frames is also a K-frame in n-Hilbert space. Also, the concept of tight K-frame in n-Hilbert space is described and some properties of its are going to be established.

Some results on singular value inequalities of compact operators in Hilbert space

We prove several singular value inequalities for sum and product of compact operators in Hilbert space. Some of our results generalize the previous inequalities for operators. Also, applications of some inequalities are given.

Some Trace Inequalities for Operators in Hilbert Spaces

Some new trace inequalities for operators in Hilbert spaces are provided. The superadditivity and monotonicity of some associated functionals are investigated and applications for power series of such operators are given. Some trace inequalities for matrices are also derived. Examples for the operator exponential and other similar functions are presented as well.

The optimal symmetric quasi-Banach range of the discrete Hilbert transform

We identify symmetric quasi-Banach range of the discrete Calderón operator and Hilbert transform acting on a symmetric quasi-Banach sequence space. As an application we present an example of optimal range in the case when the domain of those operators is the weak-$\ell_{1}$ space of sequences.

Some Identities and Inequalities for Hilbert-Schmidt Frames

In this paper we establish Parseval type identities and surprising new inequalities for Hilbert-Schmidt frames. Our results generalize and improve the remarkable results which have been obtained by Balan et al. and G{ă}vru{ţ}a.

Operators in Rigged Hilbert spaces: some spectral properties

A notion of resolvent set for an operator acting in a rigged Hilbert space $\D \subset \H\subset \D^\times$ is proposed. This set depends on a family of intermediate locally convex spaces living between $\D$ and $\D^\times$, called interspaces. Some properties of the resolvent set and of the corresponding multivalued resolvent function are derived and some examples are discussed.

Category theorems for stable operators on Hilbert spaces

We discuss the two closely related, but different concepts of weak and almost weak stability for the powers of a contraction on a separable Hilbert space. Extending Halmos' and Rohlin's theorems in ergodic theory as a model, we show that the set of all weakly stable contractions is of first category while the set of all almost weakly stable contractions is of second category and is residual. Analogous statements for unitary and isometric operators are also proved.

A Class of Sub-Hardy Hilbert Spaces Associated with Weighted Shifts

In this note we study sub-Hardy Hilbert spaces on which the the action of the operator of multiplication by the coordinate function z is assumed to be weaker than that of an isometry. We identify such operators with a class of weighted shifts. The well known results of de Branges and Beurling are deduced as corollaries .

Some more twisted Hilbert spaces

We provide three new examples of twisted Hilbert spaces by considering properties that are "close" to Hilbert. We denote them $Z(\mathcal J)$, $Z(\mathcal S^2)$ and $Z(\mathcal T_s^2)$. The first space is asymptotically Hilbertian but not weak Hilbert. On the opposite side, $Z(\mathcal S^2)$ and $Z(\mathcal T_s^2)$ are not asymptotically Hilbertian. Moreover, the space $Z(\mathcal T_s^2)$ is a HAPpy space and the technique to prove it gives a "twisted" version of a theorem of Johnson and Szankowski (Ann. of Math. 176:1987--2001, 2012). This is, we can construct a nontrivial twisted Hilbert spa…

Some variational principles over finite dimentional Hilbert spaces

In this paper a new variational approach concerning functions (continuous) over Hilbert spaces is presented.

Weak phase retrieval and phaseless reconstruction

Phase retrieval and phaseless reconstruction for Hilbert space frames is a very active area of research. Recently, it was shown that these concepts are equivalent. In this paper, we make a detailed study of a weakening of these concepts to weak phase retrieval and weak phaseless reconstruction. We will give several necessary and/or sufficient conditions for frames to have these weak properties. We will prove three surprising results: (1) Weak phaseless reconstruction is equivalent to phaseless reconstruction. I.e. It never was "weak"; (2) Weak phase retrieval is not equivalent to weak phaseles…

Hilbert-Schmidtness of some finitely generated submodules in $H^2(\mathbb{D}^2)$

A closed subspace $\mathcal{M}$ of the Hardy space $H^2(\mathbb{D}^2)$ over the bidisk is called a submodule if it is invariant under multiplication by coordinate functions $z_1$ and $z_2$. Whether every finitely generated submodule is Hilbert-Schmidt is an unsolved problem. This paper proves that every finitely generated submodule $\mathcal{M}$ containing $z_1 - φ(z_2)$ is Hilbert-Schmidt, where $φ$ is any finite Blaschke product. Some other related topics such as fringe operator and Fredholm index are also discussed.

Weakly stationary stochastic processes valued in a separable Hilbert space: Gramian-Cramér representations and applications

The spectral theory for weakly stationary processes valued in a separable Hilbert space has known renewed interest in the past decade. Here we follow earlier approaches which fully exploit the normal Hilbert module property of the time domain. The key point is to build the Gramian-Cramér representation as an isomorphic mapping from the modular spectral domain to the modular time domain. We also discuss the general Bochner theorem and provide useful results on the composition and inversion of lag-invariant linear filters. Finally, we derive the Cramér-Karhunen-Loève decomposition and harmonic f…

Equivalence of modes of convergence on reproducing kernel Hilbert spaces

Let $(X,μ)$ be a strictly-positive Borel measure space. We show that the modes of convergence in a reproducing kernel Hilbert (RKHS) space, pointwise, weak and strong are all equivalents. From this we describe some important consequences such as an association with positive operators and positive definite kernels and a compact embedding condition for a RKHS in $L^{2}(X,μ)$.

Bicomplex Linear Operators on Bicomplex Hilbert Spaces and Littlewood's Subordination Theorem

In this paper we study some basic properties of bicomplex linear operators on bicomplex Hilbert spaces. Further we discuss some applications of Hahn-Banach theorem on bicomplex Banach modules. We also introduce and discuss some bicomplex holomorphic function spaces and prove Littlewood's Subordination principle for bicomplex Hardy space.

Weak frames in Hilbert C*-modules with application in Gabor analysis

In the first part of the paper we describe the dual \ell^2(A)^{\prime} of the standard Hilbert C*-module \ell^2(A) over an arbitrary (not necessarily unital) C*-algebra A. When A is a von Neumann algebra, this enables us to construct explicitly a self-dual Hilbert A-module \ell^2_{\text{strong}}(A) that is isometrically isomorphic to \ell^2(A)^{\prime}, which contains \ell^2(A), and whose A-valued inner product extends the original inner product on \ell^2(A). This serves as a concrete realization of a general construction for Hilbert C*-modules over von Neumann algebras introduced by W. Paschk…

关于从函数空间到序列空间的Hilbert型算子

On the Hilbert-type operators acting from function spaces into sequence spaces

本文研究从函数空间到序列空间的Hilbert型算子，给出其有界性与紧性的充要条件，并在特殊情形下得到范数的精确估计。

On weaving g-frames for Hilbert spaces

Weaving frames are powerful tools in wireless sensor networks and pre-processing signals. In this paper, we introduce the concept of weaving for g-frames in Hilbert spaces. We first give some properties of weaving g-frames and present two necessary conditions in terms of frame bounds for weaving g-frames. Then we study the properties of weakly woven g-frames and give a sufficient condition for weaving g-frames. It is shown that weakly woven is equivalent to woven. Two sufficient conditions for weaving g-Riesz bases are given. And a weaving equivalent of an unconditional g-basis for weaving g-R…

A Global div-curl-Lemma for Mixed Boundary Conditions in Weak Lipschitz Domains and a Corresponding Generalized $\mathrm{A}_{0}^{*}$-$\mathrm{A}_{1}$-Lemma in Hilbert Spaces

We prove global and local versions of the so-called div-curl-lemma, a crucial result in the homogenization theory of partial differential equations, for mixed boundary conditions on bounded weak Lipschitz domains in 3D with weak Lipschitz interfaces. We will generalize our results using an abstract Hilbert space setting, which shows corresponding results to hold in arbitrary dimensions as well as for various differential operators. The crucial tools and the core of our arguments are Hilbert complexes and related compact embeddings.

More sums of Hilbert frames

In this paper we have some new results on sums of Hilbert space frames and Riesz bases. We also have a correction for some results in "S. Obeidat et al., Sums of Hilbert space frames, J. Math. Anal. Appl. 351 (2009) 579-585."

More on the metric projection onto a closed convex set in a Hilbert space

In this note, we highlight some properties of the metric projection onto a closed convex in a Hilbert space. In particular, we use some recent results on fixed points of nonexpansive potential operators.

Continuous frames in Hilbert C*-modules

In the present paper the notion of continuous frames is introduced and some results of these frames are proved. Next, we give the concept of duals of continuous frames in Hilbert C*-modules and investigate some properties of them.

弱算子拓扑下希尔伯特空间上随机指数的中心极限定理

The Central Limit Theorem for random exponents on a Hilbert space in the Weak Operator Topology

本研究针对希尔伯特空间上的随机线性连续算子，突破传统随机算子和的研究框架，探究随机指数复合的情况，证明了独立同分布线性连续算子的中心化、归一化随机指数在弱算子拓扑下满足中心极限定理，为相关算子理论研究提供了新结论。

Inequalities for Quantum f-Divergence of Trace Class Operators in Hilbert Spaces

Some inequalities for quantum f-divergence of trace class operators in Hilbert spaces are obtained. It is shown that for normalised convex functions it is nonnegative. Some upper bounds for quantum f-divergence in terms of variational and chi-distance are provided. Applications for some classes of divergence measures such as Umegaki and Tsallis relative entropies are also given.

An example of an asymptotically Hilbertian space which fails the approximation property

Following Davie's example of a Banach space failing the approximation property [D], we show how to construct a Banach space E which is asymptotically Hilbertian and fails the approximation property. Moreover, the space E is shown to be a subspace of a space with an unconditional basis which is ``almost'' a weak Hilbert space and which can be written as the direct sum of two subspaces all of whose subspaces have the approximation property.

A generalization of a half-discrete Hilbert's inequality

Considering some parameters and by means of an inequality of Hadamard, we derive general half-discrete Hilbert-type inequalities. Then we highlight some special cases.

On Orthogonal Decomposition of a Hilbert Space

In this short article we show an orthogonal decomposition of a Hilbert space as a sum of null solutions of the first derivative and the first derivative of a traceless higher order Hilbert/Sobolev space. We define orthogonal projections and see some of its properties and show case some decomposition of elementary functions as corollaries.

关于Hilbert空间中度量投影算子的严格Fréchet可微性的一些结果

Some Results on the Strict Fréchet Differentiability of the Metric Projection Operator in Hilbert Spaces

本文给出了Hilbert空间中到闭球及欧几里得空间中到二阶锥的度量投影算子严格Fréchet可微性的简洁证明与推广。

Numerical radius inequalities for Hilbert Space Operators

In this work, an improvement of Hölder-McCarty inequality is established. Based on that, several refinements of the generalized mixed Schwarz inequality are obtained. Consequently, some new numerical radius inequalities are proved. New inequalities for numerical radius of $n\times n$ matrix of Hilbert space operators are proved as well. Some refinements of some earlier results were proved in literature are also given. Some of the presented results are refined and it shown to be better than earlier results were proved in literature.

Atomic decompositions for operators in reproducing kernel Hilbert spaces

In this paper, we give some results concerning atomic decompositions for operators on reproducing kernel Hilbert spaces, using frame theory techniques. We provide also generalizations for atomic decompositions of some theorems for reproducing kernel Hilbert spaces. In particular, we obtain atomic decomposition results for operators on Bergman spaces and Fock spaces in a simple manner.

Convex-transitive characterizations of Hilbert spaces

In this paper we investigate real convex-transitive Banach spaces X, which admit a 1-dimensional bicontractive projection P on X. Various mild conditions regarding the weak topology and the geometry of the norm are provided, which guarantee that such an X is in fact isometrically a Hilbert space. The results obtained can be regarded as partial answers to the well-known Banach-Mazur rotation problem, as well as to a question posed by B. Randrianantoanina in 2002 about convex-transitive spaces.

On maximal function of discrete rough truncated Hilbert transforms

We prove the weak type (1,1) estimate for maximal function of the truncated rough Hilbert transform considered in [9] and [10]

Weighted composition operators on weak holomorphic spaces and application to weak Bloch-type spaces on the unit ball of a Hilbert space

Let $ E $ be a space of holomorphic functions on the unit ball $ B_X $ of a Banach space $ X.$ In this work, we introduce a Banach structure associated to $ E $ on the linear space $ WE(Y) $ containing $ Y$-valued holomorphic functions on $ B_X $ such that $ w \circ f \in E $ for every $ w \in W, $ a separating subspace of the dual $ Y' $ of a Banach $ Y. $ We establish the relation between the boundedness, the (weak) compactness of the weighted composition operators $ W_{ψ,φ}: f\mapsto ψ\cdot(f \circ φ) $ on $ E $ and $ \widetilde{W}_{ψ,φ}: g\mapsto ψ\cdot(g \circ φ) $ on $ WE(Y)$ via some ch…

Extended Cesàro composition operators on weak Bloch-type spaces on the unit ball of a Hilbert space

Denote by $ B_X $ the unit ball of an infinite-dimensional complex Hilbert space $ X. $ Let $ψ\in H(B_X),$ the space of all holomorphic functions on the unit ball $B_X,$ $φ\in S(B_X)$ the set of holomorphic self-maps of $B_X. $ Let $C_{ψ, φ}: \mathcal B_ν(B_X)$ (and $ \mathcal B_{ν,0}(B_X)$) $\to \mathcal B_μ(B_X) $ (and $ \mathcal B_{μ,0}(B_X)$) be weighted extended Cesàro operators induced by products of the extended Cesàro operator $ C_φ$ and integral operator $T_ψ.$ In this paper, we characterize the boundedness and compactness of $ C_{ψ,φ} $ via the estimates for the restrictions of $ ψ$ …

Hilbert $C^*$-模上伴随算子的广义极分解、弱互补性与平行和

The generalized polar decomposition, the weak complementarity and the parallel sum for adjointable operators on Hilbert $C^*$-modules

本文研究 Hilbert $C^*$-模上伴随算子，引入广义极分解工具建立弱可补算子理论，并证明了相关算子方程解的唯一性、可补性反例及平行和方程的可解性。

On typical properties of Hilbert space operators

We study the typical behavior of bounded linear operators on infinite dimensional complex separable Hilbert spaces in the norm, strong-star, strong, weak polynomial and weak topologies. In particular, we investigate typical spectral properties, the problem of unitary equivalence of typical operators, and their embeddability into C_0-semigroups. Our results provide information on the applicability of Baire category methods in the theory of Hilbert space operators.

Some Cauchy-Bunyakovsky-Schwarz Type Inequalities for Sequences of Operators in Hilbert Spaces

Some inequalities of Cauchy-Bunyakovsky-Schwarz type for sequences of bounded linear operators in Hilbert spaces and some applications are given.

On Some Numerical Radius Inequalities for Hilbert Space Operators

This article is devoted to studying some new numerical radius inequalities for Hilbert space operators. Our analysis enables us to improve an earlier bound of numerical radius due to Kittaneh.

Some generalizations of K-g-frames in Hilbert $C^{\ast}$- module

In this papers we investigate the g-frame and Bessel g-sequence related to a linear bounded operator $K$ in Hilbert $C^{\ast}$-module and we establish some results.

Some Reverses of the Continuous Triangle Inequality for Bochner Integral of Vector-Valued Functions in Complex Hilbert Spaces

Some reverses of the continuous triangle inequality for Bochner integral of vector-valued functions in complex Hilbert spaces are given. Applications for complex-valued functions are provided as well.

Ergodic theorems for affine actions of amenable groups on Hilbert space

We prove a new weak mean ergodic theorem (Theorem A) for 1-cocycles associated to weakly mixing representations of amenable groups. Let $G$ be a finitely generated, discrete, amenable group $G$ which admits a controlled Folner sequence. We use Theorem A to deduce that any affine action $G\ca^T \Cal H$ on Hilbert space with weakly mixing linear part admits a sequence of almost fixed points (Theorem B). Specializing to the case that $G$ is a finitely generated group of polynomial growth, we show that convex combinations of averages of the associated 1-cocycle over $n$-balls provide a sequence of…

Some generalizations of frames in Hilbert modules

Frames play significant role in various areas of science and engineering. In this paper, we introduce the concepts of frames for $End_{\mathcal{A}}^{\ast}(\mathcal{H, K})$ and their generalizations. Moreover, we obtain some new results for generalized frames in Hilbert modules.

Generalized Polar Decompositions for Closed Operators in Hilbert Spaces and Some Applications

We study generalized polar decompositions of densely defined, closed linear operators in Hilbert spaces and provide some applications to relatively (form) bounded and relatively (form) compact perturbations of self-adjoint, normal, and m-sectorial operators.