Mona Nabiei
This paper will initiate a study on the class of complex symmetric operators acting between two different Hilbert space. Among other things, we compute the closure of CSO with respect to the several topologies.
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In this paper, we consider composition operators on weighted Hilbert spaces of analytic functions and observe that a formula for the essential norm, give a Hilbert-Schmidt characterization and characterize the membership in Schatten-class for these operators. Also, closed range composition operators are investigated.
A well-known theorem due to R. E. Curto and N. Salinas gives a necessary and sufficient condition for the unitary equivalence of commuting tuples of bounded linear operators acting on reproducing kernel Hilbert spaces. Inspired by this theorem, we obtain a different but equivalent criterion for the unitary equivalence of operators acting on reproducing kernel Hilbert spaces. As an application, we describe the structure of intertwining operator and prove that the decomposition of Cowen-Douglas operators is unique up to unitary equivalence.
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。
In this paper we introduce and study several new Hilbert-type operators acting between the weighted Fock spaces. We provide some sufficient and necessary conditions for the boundedness and compactness of certain Hilbert-type operators from one weighted Fock space to another.
Generalized Hilbert operators acting on weighted spaces of holomorphic functions with sup-norms
本文研究了广义 Hilbert 算子在全纯函数加权 Banach 空间上的良定义、有界与紧致性，发现其有界性与紧致性等价且由测度矩可和性刻画，并分析了其核性。
Let $μ$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}_μ=(μ_{n,k})_{n,k\geq 0}$ with entries $μ_{n,k}=μ_{n+k}$, where $μ_{n}=\int_{[0,1)}t^ndμ(t)$, induces formally the operator as $$\mathcal{DH}_μ(f)(z)=\sum_{n=0}^\infty\left(\sum_{k=0}^\infty μ_{n,k}a_k\right)(n+1)z^n , z\in \mathbb{D},$$ where $f(z)=\sum_{n=0}^{\infty}a_nz^n$ is an analytic function in $\mathbb{D}$. In this paper, we characterize those positive Borel measures on $[0, 1)$ for which $\mathcal{DH}_μ$ is bounded (resp. compact) from Dirichlet spaces $\mathcal{D}_α( 0<α\leq2 )$ into $\mathcal…
The finite Hilbert transform T is a singular integral operator which maps the Zygmund space $LlogL:=LlogL(-1,1)$ continuously into $L^1:=L^1(-1,1)$. By extending the Parseval and Poincaré-Bertrand formulae to this setting, it is possible to establish an inversion result needed for solving the airfoil equation $T(f)=g$ whenever the data function $g$ lies in the range of $T$ within $L^1$ (shown to contain $LlogL$). Until now this was only known for $g$ belonging to the union of all $L^p$ spaces with $p>1$. It is established (due to a result of Stein) that $T$ cannot be extended to any domain spa…
Operators of Hilbert type acting on some spaces of analytic functions
本文研究了Hilbert类型算子在解析函数空间上的作用，探讨了其在不同空间中的有界性和紧性，并扩展了经典Hilbert算子在$H^{\infty}$上的结果。
The finite Hilbert transform acting on $L^\infty$
研究有限希尔伯特变换在 $L^\infty(-1,1)$ 上的定义与值域 $L_{\text{exp}}(-1,1)$ 的性质，揭示非可分性带来的新特征。
We compare the rate of decay of singular numbers of a given composition operator acting on various Hilbert spaces of analytic functions on the unit disk $\D$. We show that for the Hardy and Bergman spaces, our results are sharp. We also give lower and upper estimates of the singular numbers of the composition operator with symbol the ``cusp map'' and the lens maps, acting on weighted Dirichlet spaces.
Motivated by a general dilation theory for operator-valued measures, framings and bounded linear maps on operator algebras, we consider the dilation theory of the above objects with special structures. We show that every operator-valued system of imprimitivity has a dilation to a probability spectral system of imprimitivity acting on a Banach space. This completely generalizes a well-kown result which states that every frame representation of a countable group on a Hilbert space is unitarily equivalent to a subrepresentation of the left regular representation of the group. The dilated space in…
On ortho and disjointly compact operators acting to Frechet spaces
本文研究从希尔伯特空间到弗雷歇空间的算子在正交和不相交序列上的紧性性质。
Generalized Hilbert matrix operators acting on Bergman spaces
本文研究作用于单位圆盘Bergman空间$A^p$的广义Hilbert矩阵算子$\Gamma_\mu$，刻画了使其有界的测度、给出范数估计，并通过本性范数描述其紧性条件。
Generalized Hilbert matrix operators acting on weighted sequence spaces
本文引入并研究由(0,1)上正有限Borel测度诱导的广义Hilbert矩阵算子在加权序列空间上的有界性，给出了充要条件，推广了近期结果。
This article is intended to outline some the recent work by the author on the chaoticity of some specific bakward shift unbounded operators realized as differential operators acting on some Fock-Bargmann spaces and give suficient conditions on a linear unbounded densely defined chaotic shift operator $T$ acting on a Hilbert space for the operator $T+T^*$ to be chaotic where $T^*$ is its adjoint.
In this expository article, we give several examples showing how drastically different can be the behavior of operators acting on finite versus infinite dimensional Hilbert spaces. This essay is written as in such a friendly-reader to show that the situation in the infinite dimensional setting is trickier than the finite one.
We describe the norm-closures of the set $\mathfrak{C}_{\mathfrak{E}}$ of commutators of idempotent operators and the set $\mathfrak{E} - \mathfrak{E}$ of differences of idempotent operators acting on a finite-dimensional complex Hilbert space, as well as characterising the intersection of the closures of these sets with the set $\mathcal{K}(\mathcal{H})$ of compact operators acting on an infinite-dimensional, separable Hilbert space. Finally, we characterise the closures of the set $\mathfrak{C}_{\mathfrak{P}}$ of commutators of orthogonal projections and the set $\mathfrak{P} - \mathfrak{P}$…
We study two extension problems, and their interconnections: (i) extension of positive definite (p.d.) continuous functions defined on subsets in locally compact groups $G$; and (ii) (in case of Lie groups $G$) representations of the associated Lie algebras $La\left(G\right)$, i.e., representations of $La\left(G\right)$ by unbounded skew-Hermitian operators acting in a reproducing kernel Hilbert space $\mathscr{H}_{F}$ (RKHS). Our analysis is non-trivial even if $G=\mathbb{R}^{n}$, and even if $n=1$. If $G=\mathbb{R}^{n}$, (ii), we are concerned with finding systems of strongly commuting selfa…
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.
Generalized integral type Hilbert operator acting on weighted Bloch space
本文研究广义积分型Hilbert算子$\mathcal{I}_{\mu_{\alpha+1}}$在加权Bloch空间之间的有界性与紧性，并给出Bloch型空间及对数Bloch空间上的完全刻画。
We determine the structure of all bijections on the cone of positive semidefinite operators which preserve the quantum $f$-divergence for an arbitrary strictly convex function $f$ defined on the positive halfline. It turns out that any such transformation is implemented by either a unitary or an antiunitary operator.
Hilbert matrix operator acting between conformally invariant spaces
本文研究Hilbert矩阵算子从有界解析函数空间到共形不变Banach空间的作用，刻画了其到BMOA及共形不变Dirichlet空间的范数与有界性条件。
Hilbert transforms on Coxeter groups and groups acting on buildings
本文通过几何对象建立新模型，证明Coxeter群及作用于建筑的群上Hilbert变换满足Cotlar恒等式，从而得到$L_p$-有界性，并首次在具有性质(F$\mathbb{R}$)的群上实现该结果。
We show that the "eigenbundle" (localization bundle) of certain Hilbert modules over bounded symmetric domains of rank r is a "singular" vector bundle (linearly fibrered complex analytic space) which decomposes as a stratified sum of homogeneous vector bundles along a canonical stratification of length r+1. The fibres are realized in terms of representation theory on the normal space of the strata.
In this work we construct the model of a skew--selfadjoint operator with a simple spectrum acting on a Hilbert quaternion bimodule. This result is based on the Spectral Theorem for a skew--selfadjoint operator.
We analyze the "eigenbundle" (localization bundle) of certain Hilbert modules over bounded symmetric domains of rank $r,$ giving rise to complex-analytic fibre spaces which are stratified of length $r+1.$ The fibres are described in terms of Kähler geometry as line bundle sections over flag manifolds, and the metric embedding is determined by taking derivatives of reproducing kernel functions. Important examples are the determinantal ideals defined by vanishing conditions along the various strata of the stratification.
If $μ$ is a positive Borel measure on the interval $[0, 1)$, the Hankel matrix $\mathcal H_μ=(μ_{n,k})_{n,k\ge 0}$ with entries $μ_{n,k}=\int_{[0,1)}t^{n+k}\,dμ(t)$ induces formally the operator $$\mathcal{H}_μ(f)(z)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}μ_{n,k}{a_k}\right)z^n$$ on the space of all analytic functions $f(z)=\sum_{k=0}^\infty a_kz^k$, in the unit disc $\mathbb{D} $. In this paper we describe those measures $μ$ for which $\mathcal{H}_μ$ is a bounded (compact) operator from $H^p$ into $H^q$, $0<p,q<\infty $. We also characterize the measures $μ$ for which $\mathcal H_μ$ lies…
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.
On the spectrum of the symmetric tensor products of certain Hilbert-space operators
本文研究希尔伯特空间算子的对称张量积的谱，提出计算单侧加权移位与伴随算子对称及反对称张量积点谱的算法，并证明特定对称张量积的点谱仅含0。
Let $A$ be a positive bounded linear operator acting on a complex Hilbert space $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle \big)$. Let $ω_A(T)$ and ${\|T\|}_A$ denote the $A$-numerical radius and the $A$-operator seminorm of an operator $T$ acting on the semi-Hilbertian space $\big(\mathcal{H}, {\langle \cdot\mid \cdot\rangle}_A\big)$ respectively, where ${\langle x\mid y\rangle}_A := \langle Ax\mid y\rangle$ for all $x, y\in\mathcal{H}$. In this paper, we show that \begin{equation*}\label{m1} \tfrac{1}{4}\|T^{\sharp_A} T+TT^{\sharp_A}\|_A\le ω_A^2\left(T\right) \le \tfrac{1}{2}\|T^{\sh…
Generalized Hilbert operator acting on Bergman spaces
本文研究了广义积分型Hilbert算子在Bergman空间与Bloch空间之间的有界性与紧性，刻画了相关测度条件，并确定了$A^2$上的Hilbert-Schmidt类。
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…
We investigate the spectrum and fine spectra of the finite Hilbert transform acting on rearrangement invariant spaces over $(-1,1)$ with non-trivial Boyd indices, thereby extending Widom's results for $L^p$ spaces. In the case when these indices coincide, a full description of the spectrum and fine spectra is given.
A distance between operators acting in different Hilbert spaces and operator convergence
本文介绍了拟酉等价概念，在可能作用于不同Hilbert空间的自伴算子空间上首次定义了若干缺乏部分度量性质的“距离”，为后续研究奠定基础。
$A$-approximate point spectrum of $A$-bounded operators in semi-Hilbertian spaces
该研究聚焦复半希尔伯特空间上的算子谱理论，刻画A-有界算子的A-近似点谱，分析A^(1/2)-伴随算子张量积的对应谱性质，并推导A-正规算子的相关性质，丰富了半希尔伯特空间算子理论的内容。
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。
Let $μ$ be a positive Borel measure on the interval [0,1). The Hankel matrix $\mathcal{H}_μ= (μ_{n,k})_{n,k\geq0}$ with entries $μ_{n,k}= μ_{n+k}$, where $μ_n=\int_{ [0,1)}t^ndμ(t)$, induces formally the operator $$\mathcal{DH}_μ(f)(z)=\sum_{n=0}^\infty (\sum_{k=0}^\infty μ_{n,k}a_k)(n+1)z^n$$ on the space of all analytic function $f(z)=\sum_{k=0}^ \infty a_k z^n$ in the unit disc $\mathbb{D}$. We characterize those positive Borel measures on $[0,1)$ such that $\mathcal{DH}_μ(f)(z)= \int_{[0,1)} \frac{f(t)}{(1-tz)^2} dμ(t)$ for all in Hardy spaces $H^p(0<p<\infty)$, and among them we describe …
It is a longstanding problem whether every contractible Banach algebra is necessarily finite-dimensional. In this note, we confirm this for Banach algebras acting on Banach spaces with the uniform approximation property. This generalizes a result of Paulsen and Smith who proved the same for Banach algebras acting on Hilbert spaces.
In a previous paper, the authors introduced the idea of a symmetric pair of operators as a way to compute self-adjoint extensions of symmetric operators. In brief, a symmetric pair consists of two densely defined linear operators $A$ and $B$, with $A \subseteq B^\star$ and $B \subseteq A^\star$. In this paper, we will show by example that symmetric pairs may be used to deduce closability of operators and sometimes even compute adjoints. In particular, we prove that the Malliavin derivative and Skorokhod integral of stochastic calculus are closable, and the closures are mutually adjoint. We als…
On $C$-Symmetric and $C$-Self-adjoint Unbounded Operators on Hilbert Space
本文研究希尔伯特空间上关于共轭$C$的$C$-对称与$C$-自伴无界算子，给出了$C$-自伴扩张的完全描述、基于拟解析向量的判据及极分解刻画。
New inequalities for the $A$-numerical radius of the products and sums of operators acting on a semi-Hilbert space, i.e. a space generated by a positive semidefinite operator $A$, are established. In particular, it is proved for operators $T$ and $S,$ having $A$-adjoint, that $$ ω_A(TS) \leq \frac{1}{2}ω_A(ST)+\frac{1}{4}\Big(\|T\|_A\|S\|_A+\|TS\|_A\Big),$$ where $ω_A(T)$ and $\|T\|_A$ denote the $A$-numerical radius and the $A$-operator seminorm of an operator $T$.
Let X be a real Banach space. We prove that the existence of an injective, positive, symmetric and not strictly singular operator from X into its dual implies that either X admits an equivalent Hilbertian norm or it contains a nontrivially complemented subspace which is isomorphic to a Hilbert space. We also treat the non-symmetric case.
In this paper we characterize Toeplitz matrices with entries in the space of bounded operators on Hilbert spaces $\mathcal{B}(H)$ which define bounded operators acting on $\ell^2(H)$ and use it to get the description of the right Schur multipliers acting on $\ell^2(H)$ in terms of certain operator-valued measures.
On products of symmetries acting on Hilbert spaces
本文研究复可分希尔伯特空间上酉算子群中对称性乘积构成的子集$\text{Sym}_k(\mathcal{H})$，刻画有限维下$\text{Sym}_3(\mathcal{H})$的特征值性质，并探讨$k\le3$时对称性轨道与酉轨道的等价条件。
Generalized Hilbert Operator Acting on Hardy Spaces
本文研究由正 Borel 测度诱导的广义 Hilbert 算子在 Hardy 空间之间的有界性、紧性及本质范数。
Properties of reproducing kernel Hilbert spaces of a group action
本文在紧豪斯多夫空间$X$上引入等价关系并建立其三个等价定义，探讨了$X$的等价类与群的关系稳定子群的对应关系，研究了群作用的再生核希尔伯特空间的性质。
In general, it is a non trivial task to determine the adjoint $S^*$ of an unbounded operator $S$ acting between two Hilbert spaces. We provide necessary and sufficient conditions for a given operator $T$ to be identical with $S^*$. In our considerations, a central role is played by the operator matrix $M_{S,T}=\left(\begin{array}{cc} I & -T\\ S & I\end{array}\right)$. Our approach has several consequences such as characterizations of closed, normal, skew- and selfadjoint, unitary and orthogonal projection operators in real or complex Hilbert spaces. We also give a self-contained proof of the f…
Königs maps and commutants of composition operators on the Hardy-Hilbert space of Dirichlet series
本文研究了Dirichlet级数Hardy-Hilbert空间上的有界复合算子，通过构造Königs映射推导出其谱与动力学性质，并探讨了一大类符号下复合算子是否具有极小换子。
A Derivative-Hilbert operator acting on BMOA space
本文刻画了使导数-希尔伯特算子有界于BMOA空间的测度条件，并研究了从α-Bloch空间到BMOA空间的类似问题。
In this paper, we shall characterize the components of the polar decomposition for an arbitrary $J$-unitary operator in a Hilbert space. This characterization has a quite different structure as that for complex symmetric and complex skew-symmetric operators. It is also shown that for a $J$-imaginary closed symmetric operator in a Hilbert space there exists a $J$-imaginary self-adjoint extension in a possibly larger Hilbert space (a linear operator $A$ in a Hilbert space $H$ is said to be $J$-imaginary if $f\in D(A)$ implies $Jf\in D(A)$ and $AJf = -JAf$, where $J$ is a conjugation on $H$).
In this paper, we study the representation of orthogonally additive mappings acting on Hilbert $C^*$-modules and Hilbert $H^*$-modules. One of our main results shows that every continuous orthogonally additive mapping $f$ from a Hilbert module $W$ over $\K(\H)$ or $\H§(\H)$ to a complex normed space is of the form $f(x)=T(x)+Φ(< x, x >)$ for all $x\in W$, where $T$ is a continuous additive mapping, and $Φ$ is a continuous linear mapping.
Convergence of operators acting on a given Hilbert space is an old and well studied topic in operator theory. The idea of introducing a related notion for operators acting on arying spaces is natural. However, it seems that the first results in this direction have been obtained only recently, to the best of our knowledge. Here we consider sectorial operators on scales of Hilbert spaces. We define a notion of convergence that generalises convergence of the resolvents in operator norm to the case when the operators act on different spaces and show that this kind of convergence is compatible with…
For a shift operator $T$ with finite multiplicity acting on a separable infinite dimensional Hilbert space we represent its nearly $T^{-1}$ invariant subspaces in terms of invariant subspaces under the backward shift. Going further, given any finite Blaschke product $B$, we give a description of the nearly $T_{B}^{-1}$ invariant subspaces for the operator $T_B$ of multiplication by $B$ in a scale of Dirichlet-type spaces.
An operator $T$ acting on a Hilbert space is called $(α,β)$-normal ($0\leq α\leq 1\leq β$) if \begin{equation*} α^{2}T^{\ast }T\leq TT^{\ast}\leq β^{2}T^{\ast}T. \end{equation*} In this paper we establish various inequalities between the operator norm and its numerical radius of $(α,β)$-normal operators in Hilbert spaces. For this purpose, we employ some classical inequalities for vectors in inner product spaces.
We prove $L^p$-bounds for the bilinear Hilbert transform acting on functions valued in intermediate UMD spaces. Such bounds were previously unknown for UMD spaces that are not Banach lattices. Our proof relies on bounds on embeddings from Bochner spaces $L^p(\mathbb{R};X)$ into outer Lebesgue spaces on the time-frequency-scale space $\mathbb{R}^3_+$.
On the Hilbert matrix operator: a brief survey
本文是一篇关于希尔伯特矩阵算子的简要综述，梳理了其作为解析函数空间与序列空间上算子的最新研究进展，为算子矩阵理论相关研究者提供参考。
Universality arising from invertible weighted composition operators
本文研究了可逆加权双曲复合算子的通用性，证明了其特征值平移在Hardy和加权Bergman空间上具有通用性，并讨论了Banach空间情形的推论。
We show that the Hilbert space compression of any finite dimensional CAT(0) cube complex is 1 and deduce that any discrete group acting properly, co-compactly on a CAT(0) cube complex is exact. The class of groups covered by this theorem includes free groups, finitely generated Coxeter groups, finitely generated right angled Artin groups, finitely presented groups satisfying the B(4)-T(4) small cancellation condition and all those word-hyperbolic groups satisfying the B(6) condition. Another family of examples is provided by certain canonical surgeries defined by link diagrams.
In topological equivalence, a bounded linear operator between Banach spaces - we focus on the case of Hilbert spaces - is viewed as only acting linearly and continuously between them qua different spaces with the structure of linear topological space. For instance, invertible operators in Banach spaces (that is, isomorphisms among them) will make up one equivalence class for each class of isomorphic spaces. On the other hand, compact and non-compact operators, or operators with or without a kernel, clearly will not. We make some crucial steps towards describing invariants that will characteriz…
If $g$ is an analytic function in the unit disc $\D $ we consider the generalized Hilbert operator $\hg$ defined by {equation*}\label{H-g} \mathcal{H}_g(f)(z)=\int_0^1f(t)g'(tz)\,dt. {equation*} We study these operators acting on classical spaces of analytic functions in $\D $. More precisely, we address the question of characterizing the functions $g$ for which the operator $\hg $ is bounded (compact) on the Hardy spaces $H^p$, on the weighted Bergman spaces $A^p_α$ or on the spaces of Dirichlet type $\mathcal D^p_α$.
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