An Operator Hilbert space without the operator approximation property

On the $\text{OUMD}$ property for the column Hilbert space $C$

The operator space $\text{OUMD}$ property was introduced by Pisier in the context of vector-valued noncommutative $L_p$-spaces. It is an open problem whether the column Hilbert space has this property. Based on some complex interpolation techniques, we are able to relate this problem to the study of the $\text{OUMD}_q$ property for non-commutative $L_p$-spaces.

Note on the Kato property of sectorial forms

We characterise the Kato property of a sectorial form $\mathfrak{a}$, defined on a Hilbert space $V$, with respect to a larger Hilbert space $H$ in terms of two bounded, selfadjoint operators $T$ and $Q$ determined by the imaginary part of $\mathfrak{a}$ and the embedding of $V$ into $H$, respectively. As a consequence, we show that if a bounded selfadjoint operator $T$ on a Hilbert space $V$ is in the Schatten class $S_p(V)$ ($p\geq 1$), then the associated form $\mathfrak{a}_T(\cdot, \cdot) := \langle (I+iT)\cdot ,\cdot\rangle_V$ has the Kato property with respect to every Hilbert space $H$ …

Approximation of the Image of the Hilbert-Schmidt Integral Operator

In this paper an approximation of the image of the closed ball of the space $L_p$ $(p>1)$ centered at the origin with radius $r$ under Hilbert-Schmidt integral operator $F(\cdot):L_p\rightarrow L_q$ $\displaystyle \left(\frac{1}{p}+\frac{1}{q}=1\right)$ is presented. An error estimation for given approximation is obtained.

Non-Hilbert Banach spaces with the self-extension property

In 1992, Kiendi, Adamy and Stelzner investigated under which conditions a certain type of function constituted a Lyapunov function for some time-invariant linear system. Six years later, it was obtained that this property holds if and only if the Banach space enjoys the self-extension property. However, the knowledge of these spaces needed to be extended in order to make useful this characterization, since there were little information on which classic Banach spaces satisfy this property and its relations with other classic properties. We present the self-extension property in a wider frame, r…

Approximation of Besov vectors by Paley-Wiener vectors in Hilbert spaces

We develop an approximation theory in Hilbert spaces that generalizes the classical theory of approximation by entire functions of exponential type. The results advance harmonic analysis on manifolds and graphs, thus facilitating data representation, compression, denoising and visualization. These tasks are of great importance to machine learning, complex data analysis and computer vision.

再生核 Hilbert 空间中的概率逼近

On the Probabilistic Approximation in Reproducing Kernel Hilbert Spaces

本文研究了再生核 Hilbert 空间中的概率函数逼近问题，证明了优化器的存在唯一性，推广了表示定理，并揭示了其在有限支撑与无限支撑下与测度量化和采样理论的联系。

Rokhlin Property for Group Actions on Hilbert $C^*$-modules

We introduce Rokhlin properties for certain discrete group actions on $C^*$-correspondences as well as on Hilbert bimodules and analyze them. It turns out that the group actions on any $C^*$-correspondence $E$ with Rokhlin property induces group actions on the associated $C^*$-algebra $\mathcal O_E$ with Rokhlin property and the group actions on any Hilbert bimodule with Rokhlin property induces group actions on the linking algebra with Rokhlin property. Permanence properties of several notions such as nuclear dimension and $\mathcal D$-absorbing property with respect to crossed product of Hil…

Approximation of fixed points of nonexpansive mappings and quasinonexpansive mappings in a Hilbert space

In this paper, we give a simple proof and some generalizations of results in Falset, Llorens-Fuster, Marino, and Rugiano (2016).

The proportional UAP characterizes weak Hilbert spaces

We prove that a Banach space has the uniform approximation property with proportional growth of the uniformity function iff it is a weak Hilbert space.

The Bishop-Phelps-Bollobás properties in complex Hilbert spaces

In this paper we consider a stronger property than the Bishop-Phelps-Bollobás property for various classes of operators on a complex Hilbert space. The Bishop-Phelps-Bollobás {\it point} property for some class $\mathcal{A} \subset \mathcal{L}(H)$ says that if one starts with a norm one operator $T$ belonging to $\mathcal{A}$, which almost attains its norm at some norm one vector $x_0$, then there is a new operator $S$, belonging to the same class $\mathcal{A}$, which is close to $T$ and attains its norm at the same vector $x_0$. We study it for classical operators on a complex Hilbert spaces …

Approximation of the Inverse Frame Operator and Stability of Hilbert$-$Schmidt Frames

In this paper, we study the Hilbert$-$Schmidt frame (HS-frame) theory for separable Hilbert spaces. We first present some characterizations of HS-frames and prove that HS-frames share many important properties with frames. Then, we show how the inverse of the HS-frame operator can be approximated using finite-dimensional methods. Finally, we present a classical perturbation result and prove that HS-frames are stable under small perturbations.

Quasianalytic n-tuples of Hilbert space operators

In this paper a systematic study of unitary asymptotes of commuting $n$-tuples of general Hilbert space operators is initiated. Special emphasis is put on the study of the quasianalicity property.

Global solutions of approximation problems in Hilbert spaces

We study three well-known minimization problems in Hilbert spaces: the weighted least squares problem and the related problems of abstract splines and smoothing. In each case we analyze the solvability of the problem for every point of the Hilbert space in the corresponding data set, the existence of an operator that maps each data point to its solution in a linear and continuous way and the solvability of the associated operator problem in a fixed p-Schatten norm.

Real analytic approximations which almost preserve Lipschitz constants of functions defined on the Hilbert space

Let $X$ be a separable real Hilbert space. We show that for every Lipschitz function $f:X\rightarrow\mathbb{R}$, and for every $ε>0$, there exists a Lipschitz, real analytic function $g:X\rightarrow\mathbb{R}$ such that $|f(x)-g(x)|\leq ε$ and $\textrm{Lip}(g)\leq \textrm{Lip}(f)+ε$.

Hilbert modules over a planar algebra and the Haagerup property

Given a subfactor planar algebra P and a Hilbert P-module of lowest weight 0 we build a bimodule over the symmetric enveloping inclusion associated to P. As an application we prove diagrammatically that the Temperley-Lieb-Jones standard invariants have the Haagerup property. This provides a new proof of a result due to Popa and Vaes.

Hilbert空间上算子的Schröder-Bernstein性质

The Schröder-Bernstein property for operators on Hilbert spaces

本文证明了配备正规算子的Hilbert空间的完备理论具有Schröder-Bernstein性质，回答了相关提问，并对无界自伴算子证明了类似结论。

Banach spaces with the $2$-summing property

A Banach space $X$ has the $2$-summing property if the norm of every linear operator from $X$ to a Hilbert space is equal to the $2$-summing norm of the operator. Up to a point, the theory of spaces which have this property is independent of the scalar field: the property is self-dual and any space with the property is a finite dimensional space of maximal distance to the Hilbert space of the same dimension. In the case of real scalars only the real line and real $\ell_\infty^2$ have the $2$-summing property. In the complex case there are more examples; e.g., all subspaces of complex $\ell_\in…

Which de Branges-Rovnyak spaces have complete Nevanlinna-Pick property?

We characterize the de Branges-Rovnyak spaces with complete Nevanlinna-Pick property. Our method relies on the general theory of reproducing kernel Hilbert spaces.

Real analytic approximation of Lipschitz functions on Hilbert space and other Banach spaces

Let $X$ be a separable Banach space with a separating polynomial. We show that there exists $C\geq 1$ (depending only on $X$) such that for every Lipschitz function $f:X\rightarrow\mathbb{R}$, and every $ε>0$, there exists a Lipschitz, real analytic function $g:X\rightarrow\mathbb{R}$ such that $|f(x)-g(x)|\leq ε$ and $\textrm{Lip}(g)\leq C\textrm{Lip}(f)$. This result is new even in the case when $X$ is a Hilbert space. Furthermore, in the Hilbertian case we also show that $C$ can be assumed to be any number greater than 1.

Inequalities of Jackson-Stechkin type for elements of Hilbert space (in Russian)

In this paper we introduced a new characteristics of the elements of a Hilbert space - generalized moduli of continuity $ω_φ(x;L_{p,V}([0,δ]))$ and obtain new exact inequalities of Jackson - Stechkin type with these moduli of continuity for the approximation of elements of a Hilbert space. These results include many well-known inequalities for approximation of periodic functions by trigonometric polynomials, approximation of non-periodic functions by entire functions of exponential type, similar results for almost periodic functions, and others. A number of results is new even in these classic…

The Compact Approximation Property does not imply the Approximation Property

It is shown how to construct, given a Banach space which does not have the approximation property, another Banach space which does not have the approximation property but which does have the compact approximation property.

Stampacchia's property, self-duality and orthogonality relations

We show that if the conclusion of the well known Stampacchia Theorem, on variational inequalities, holds on a Banach space X, then X is isomorphic to a Hilbert space. Motivated by this we obtain a relevant result concerning self-dual Banach spaces and investigate some connections between existing notions of orthogonality and self-duality. Moreover, we revisit the notion of the cosine of a linear operator and show that it can be used to characterize Hilbert space structure. Finally, we present some consequences of our results to quadratic forms and to evolution triples.

Hilbert空间中框架与基的对称逼近

Symmetric approximation of frames and bases in Hilbert spaces

本文研究Hilbert空间中框架和基的对称逼近问题，利用算子(P-|F|)的Hilbert-Schmidt性质刻画归一化紧框架和标准正交基的二次最近逼近的存在性与唯一性，结果可应用于小波理论。

Separable morphisms of operator Hilbert systems, Pietsch factorizations and entanglement breaking maps

In this paper we investigate operator Hilbert systems and their separable morphisms. We prove that the operator Hilbert space of Pisier is an operator system, which possesses the self-duality property. It is established a link between unital positive maps and Pietch factorizations, which allows us to describe all separable morphisms from an abelian C*-algebra to an operator Hilbert system. Finally, we prove a key property of entanglement breaking maps that involves operator Hilbert systems.

An example of a Banach Space with a Subsymmetric Basis, which has the Hereditarily Approximation Property

W.B. Johnson has constructed a series of Banach spaces non isomorphic to the Hilbert one that have the hereditarily approximation property (shortly hereditarily AP): all their subspaces also have the AP. All these examples were ''sufficiently'' non-symmetric and this fact allows Johnson to ask: whether there exists any Banach space $X$ with symmetric (or, at least, subsymmetric) basis, distinct from the Hilbert space such that each its subspace has the AP? In this paper is shown that there is a Banach space with a subsymmetric basis (non-equivalent to any symmetric one), which enjoys the hered…

Symmetric multilinear forms on Hilbert spaces: where do they attain their norm?

We characterize the sets of norm one vectors $\mathbf{x}_1,\ldots,\mathbf{x}_k$ in a Hilbert space $\mathcal H$ such that there exists a $k$-linear symmetric form attaining its norm at $(\textbf{x}_1,\ldots,\mathbf{x}_k)$. We prove that in the bilinear case, any two vectors satisfy this property. However, for $k\ge 3$ only collinear vectors satisfy this property in the complex case, while in the real case this is equivalent to $\mathbf{x}_1,\ldots,\mathbf{x}_k$ spanning a subspace of dimension at most 2. We use these results to obtain some applications to symmetric multilinear forms, symmetric…

Inversion Theorem for Bilinear Hilbert Transform

An approximation result for the bilinear Hilbert transform is proved and used for the inversion of the bilinear Hilbert transform. Also, p-Lebesgue points $(p\geq 1)$ are analyzed.

A Gaussian Average Property for Banach Spaces

In this paper we investigate a Gaussian average property of Banach spaces. This property is weaker than the Gordon Lewis property but closely related to this and other unconditional structures. It is also shown that this property implies that certain Hilbert space valued operators defined on subspaces of the given space can be extended.

Fractional Riesz-Hilbert transforms and fractional monogenic signals

The fractional Hilbert transforms plays an important role in optics and signal processing. In particular the analytic signal proposed by Gabor has as a key component the Hilbert transform. The higher dimensional Hilbert transform is the Riesz-Hilbert transform which was used by Felsberg and Sommer to construct the monogenic signal. We will construct fractional and quaternionic fractional Riesz-Hilbert transforms based on a eigenvalue decomposition. We will prove properties of these transformations such as shift and scale invariance, orthogonality and the semigroup property. Based on the fracti…

P-Operators on Hilbert Spaces

A real square matrix $A$ is called a P-matrix if all its principal minors are positive. Using the sign non-reversal property of matrices, the notion of P-matrix has been recently extended by Kannan and Sivakumar to infinite-dimensional Banach spaces relative to a given Schauder basis. Motivated by their work, we discuss P-operators on separable real Hilbert spaces. We also investigate P-operators relative to various orthonormal bases.

典型 Hilbert 空间算子的谱

Spectra of typical Hilbert space operators

本文研究了可分 Hilbert 空间上有界线性算子的典型性质，证明了具有非空连续谱是典型性质，并刻画了点谱相关算子集合的稠密性与闭包。

The Lipschitz bounded approximation property for operator ideals

In this article, we introduce the Lipschitz bounded approximation property for operator ideals. With this notion, we extend the original work done by Godefroy and Kalton and give some partial answers on the equivalence between the bounded approximation property and the Lipschitz bounded approximation property based on an arbitrary operator ideal. Furthermore, we investigate the three space problems on the preceding bounded approximation properties.

Stereotype approximation property for the group algebra ${\mathcal C}^\star(G)$ of measures

The stereotype approximation property is formally a stronger condition than the classical approximation property, and because of that the question which spaces possess the stereotype approximation property is quite difficult. In this paper we show that the group algebra ${\mathcal C}^\star(G)$ of measures on a locally compact grop $G$ always has this property.

The approximation property and Lipschitz mappings on Banach spaces

We present an overview to the approximation property, paying especial attention to the recent results relating the approximation property to ideals of linear operators and Lipschitz ideals. We complete the paper with some new results on approximation of Lipschitz mappings and their relation to linear operator ideals.

Trotter-Kato Approximations of Impulsive Neutral SPDEs in Hilbert Spaces

This paper studies a class of impulsive neutral stochastic partial differential equations in real Hilbert spaces. The main goal here is to consider the Trotter-Kato approximations of mild solutions of such equations in the $p$th-mean ($p\geq2$). As an application, a classical limit theorem on the dependence of such equations on a parameter is obtained. The novelty of this paper is that the combination of this approximating system and such equations has not been considered before.

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.

关于可离散化希尔伯特空间中算子压缩半群的逼近

On the Approximation of Contractive Semigroups of Operators in Discretizable Hilbert Spaces

研究可离散化希尔伯特空间中由逼近方案诱导的离散压缩半群的性质，并应用于量子动力学半群的Heisenberg表示。

可分离Hilbert空间上Hilbert-Schmidt算子的广义秩约束逼近及其应用

Generalised Rank-Constrained Approximations of Hilbert-Schmidt Operators on Separable Hilbert Spaces and Applications

本文针对有界算子B、C和Hilbert-Schmidt算子M，求解了秩约束逼近问题min{||M-BXC||_{L_2}: dim ran X ≤ r}，推广了矩阵情形的结果，并分析了解的不连续性、最小性性质，最后应用于信号处理、降秩回归和线性算子学习。

Approximation of Hilbert-valued Gaussians on Dirichlet structures

We introduce a framework to derive quantitative central limit theorems in the context of non-linear approximation of Gaussian random variables taking values in a separable Hilbert space. In particular, our method provides an alternative to the usual (non-quantitative) finite dimensional distribution convergence and tightness argument for proving functional convergence of stochastic processes. We also derive four moments bounds for Hilbert-valued random variables with possibly infinite chaos expansion, which include, as special cases, all finite-dimensional four moments results for Gaussian app…

Frame analysis and approximation in reproducing kernel Hilbert spaces

We consider frames F in a given Hilbert space, and we show that every F may be obtained in a constructive way from a reproducing kernel and an orthonormal basis in an ambient Hilbert space. The construction is operator-theoretic, building on a geometric formula for the analysis operator defined from F. Our focus is on the infinite-dimensional case where a priori estimates play a central role, and we extend a number of results which are known so far only in finite dimensions. We further show how this approach may be used both in constructing useful frames in analysis and in applications, and in…

On approximation properties of Pimsner algebras and crossed products by Hilbert bimodules

Let X be a Hilbert bimodule over a C*-algebra A and $O_X= A \rtimes_X \Z$. Using a finite section method we construct a sequence of completely positive contractions factoring through matrix algebras over A which act on $s_ξ s_η^*$ as Schur multipliers converging to the identity. This shows immediately that for a finitely generated X the algebra $O_X$ inherits any standard approximation property such as nuclearity, exactness, CBAP or OAP from A. We generalise this to certain general Pimsner algebras by proving semi-splitness of the Toeplitz extension under certain conditions and discuss some ex…

Hilbert-Poisson 空间上的函数型高斯逼近

Functional Gaussian approximations on Hilbert-Poisson spaces

本文在 Hilbert-Poisson 空间上建立函数型 Stein-Malliavin 方法，结合无穷维可交换对与 Gamma 演算，给出高斯随机元逼近的定量中心极限定理和四阶矩界，并应用于 Poisson 过程的 Brown 逼近及随机几何图边计数统计量。

Invariant Hermitean Metric Generated By Reproducing Kernel Hilbert Spaces

For a Reproducing Kernel Hilbert Space on a complex domain we give a formula that describes the Hermitean metrics on the domain which are pull-backs of some metric on the (dual of) the RKHS via the evaluation map. Then we consider the question when such metrics are invariant with respect to the group of automorphisms of the domain. First we approach the problem by considering a stronger property, demanding that the original metric on the (dual of) the RKHS is invariant with respect to all (adjoints of) composition operators, induced by automorphisms. However, we show that only the trivial metr…

Hilbert-Schmidt 算子与框架——分类、乘子逼近及算法

Hilbert-Schmidt Operators and Frames - Classification, Approximation by Multipliers and Algorithms

本文利用框架给出Hilbert-Schmidt算子的简单判别准则，通过张量积构造Bessel序列、框架和Riesz基，并设计算法用框架乘子最佳逼近任意矩阵。

Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds

We prove that every continuous mapping from a separable infinite-dimensional Hilbert space $X$ into $\mathbb{R}^{m}$ can be uniformly approximated by $C^\infty$ smooth mappings {\em with no critical points}. This kind of result can be regarded as a sort of very strong approximate version of the Morse-Sard theorem. Some consequences of the main theorem are as follows. Every two disjoint closed subsets of $X$ can be separated by a one-codimensional smooth manifold which is a level set of a smooth function with no critical points; this fact may be viewed as a nonlinear analogue of the geometrical…

Amenable representations and dynamics of the unit sphere in an infinite-dimensional Hilbert space

We establish a close link between the amenability of a unitary representation $π$ of a group $G$ (in the sense of Bekka) and the concentration property (in the sense of V. Milman) of the corresponding dynamical system $(\s_π,G)$, where $\s_\H$ is the unit sphere the Hilbert space of representation. We prove that $π$ is amenable if and only if either $π$ contains a finite-dimensional subrepresentation or the maximal uniform compactification of $\s_π$ has a $G$-fixed point. Equivalently, the latter means that the $G$-space $(\s_π,G)$ has the concentration property: every finite cover of the sphe…

Trotter-Kato approximations of semilinear stochastic evolution equations in Hilbert spaces

Motivated by the work of T.E. Govindan in [5,8,9], this paper is concerned with a more general semilinear stochastic evolution equation. The difference between the equations considered in this paper and the previous one is that it makes some changes to the nonlinear function in random integral, which also depends on the probability distribution of stochastic process at that time. First, this paper considers the existence and uniqueness of mild solutions for such equations. Furthermore, Trotter-Kato approximation system is introduced for the mild solutions, and the weak convergence of induced p…

A short note on compact embeddings of reproducing kernel Hilbert spaces in $L^2$ for infinite-variate function approximation

This note consists of two largely independent parts. In the first part we give conditions on the kernel $k: Ω\times Ω\rightarrow \mathbb{R}$ of a reproducing kernel Hilbert space $H$ continuously embedded via the identity mapping into $L^2(Ω, μ),$ which are equivalent to the fact that $H$ is even compactly embedded into $L^2(Ω, μ).$ In the second part we consider a scenario from infinite-variate $L^2$-approximation. Suppose that the embedding of a reproducing kernel Hilbert space of univariate functions with reproducing kernel $1+k$ into $L^2(Ω, μ)$ is compact. We provide a simple criterion fo…

Fixed points for bounded orbits in Hilbert spaces

Consider the following property of a topological group G: every continuous affine G-action on a Hilbert space with a bounded orbit has a fixed point. We prove that this property characterizes amenability for locally compact sigma-compact groups (e.g. countable groups). Along the way, we introduce a "moderate" variant of the classical induction of representations and we generalize the Gaboriau--Lyons theorem to prove that any non-amenable locally compact group admits a probabilistic variant of discrete free subgroups. This leads to the "measure-theoretic solution" to the von Neumann problem for…

Orthonormal bases of Hilbert spaces

I prove that a Hilbert space has the property that each of its dense (not necessarily closed) subspaces contains an orthoormal basis if and only if it is separable.

A property of C_p[0,1]

We prove that for every finite dimensional compact metric space $X$ there is an open continuous linear surjection from $C_p[0,1]$ onto $C_p(X)$. The proof makes use of embeddings introduced by Kolmogorov and Sternfeld in connection with Hilbert's 13th problem.

Duality of uniform approximation property in operator spaces

The duality of uniform approximation property for Banach spaces is well known. In this note, we establish, under the assumption of local reflexivity, the duality of uniform approximation property in the category of operator spaces.

Stable polynomial division and essential normality of graded Hilbert modules

The purpose of this paper is to initiate a new attack on Arveson's resistant conjecture, that all graded submodules of the $d$-shift Hilbert module $H^2$ are essentially normal. We introduce the stable division property for modules (and ideals): a normed module $M$ over the ring of polynomials in $d$ variables has the stable division property if it has a generating set $\{f_1, ..., f_k\}$ such that every $h \in M$ can be written as $h = \sum_i a_i f_i$ for some polynomials $a_i$ such that $\sum \|a_i f_i\| \leq C\|h\|$. We show that certain classes of modules have this property, and that the s…

Stability of group relations under small Hilbert-Schmidt perturbations

If matrices almost satisfying a group relation are close to matrices exactly satisfying the relation, then we say that a group is matricially stable. Here "almost" and "close" are in terms of the Hilbert-Schmidt norm. Using tracial 2-norm on $II_1$-factors we similarly define $II_1$-factor stability for groups. Our main result is that all 1-relator groups with non-trivial center are $II_{1}$-factor stable. Many of them are also matricially stable and RFD. For amenable groups we give a complete characterization of matricial stability in terms of the following approximation property for characte…