Chafiq Benhida, Emmanuel Fricain, Dan Timotin
Using the Sz.-Nagy--Foias theory of contractions, we obtain general results about reducibility for a class of completely nonunitary contractions. These are applied to certain truncated Toeplitz operators, previously considered by Li--Yang--Lu and Gu. In particular, a negative answer is given to a conjecture stated by the latter.
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We investigate different possiblities of subspaces of the space $\ell_{\infty}$ in terms of whether the subspaces are polyhedral or not. We further study finite-dimensional subspaces of $\ell_{\infty}$ which are of the form $\ell_\infty^n$ form some $ n \geq 2.$ As an application of the results we compute the number of extreme contractions for a class of the space of bounded linear operators. In particular we find the number of extreme contractions of $\mathbb{L}(\mathbb{X}, \ell_{\infty}^n),$ where $\mathbb{X}$ is a finite-dimensional polyhedral space.
The main result of the paper is that a system of invariant subspaces of a (completely non-unitary) Hilbert space contraction $T$ with finite defects (rank$(I-T^*T)<\infty$, rank$(I-TT^*)<\infty$) is an unconditional basis (Riesz basis) if and only if it is uniformly minimal. Results of such type are quite well known: for a system of eigenspaces of a contraction with defects $1-1$ it is simply the famous Carleson interpolation theorem. For general invariant subspaces of operators with defects $1-1$ such theorem was proved by V. I. Vasyunin. Then partial results for the case of finite defects we…
Given a Polish topology $τ$ on ${\mathcal{B}_{1}(X)}$, the set of all contraction operators on $X=\ell_p$, $1\le p<\infty$ or $X=c_0$, we prove several results related to the following question: does a typical $T\in {\mathcal{B}_{1}(X)}$ in the Baire Category sense has a non-trivial invariant subspace? In other words, is there a dense $G_δ$ set $\mathcal G\subseteq ({\mathcal{B}_{1}(X)},τ)$ such that every $T\in\mathcal G$ has a non-trivial invariant subspace? We mostly focus on the Strong Operator Topology and the Strong$^*$ Operator Topology.
Non-stable subnormal contractions have nontrivial hyperinvariant subspaces
研究非稳定纯次正规收缩算子的超不变子空间存在性，证明存在奇异内函数θ使得θ(T)的范围不密集，从而得出算子具有非平凡超不变子空间。
Theory of q-commuting contractions : joint reducing subspaces and orthogonal decompositions
本文将交换压缩的典范分解结果推广至 $Q$-交换压缩，主要研究 $Q$ 为酉算子族时的 $Q$-交换与双重 $Q$-交换压缩的联合约化子空间与正交分解。
A commuting triple of Hilbert space operators $(A,B,P)$, for which the closed tetrablock $\overline{\mathbb E}$ is a spectral set, is called a \textit{tetrablock-contraction} or simply an $\mathbb E$-\textit{contraction}, where \[ \mathbb E=\{(x_1,x_2,x_3)\in \mathbb C^3:\, 1-x_1z-x_2w+x_3zw \neq 0 \quad \text{ whenever } \; |z|\leq 1, \; \; |w|\leq 1 \} \subset \mathbb C^3, \] is a polynomially convex domain which is naturally associated with the $μ$-synthesis problem. By applications of the theory of $\mathbb E$-contractions, we obtain several results on decompositions of contractions.
It is known that pure row contractions with one-dimensional defect spaces can be classified up to unitary equivalence by compressions of the standard $d$-shift acting on the full Fock space. Upon settling for a softer relation than unitary equivalence, we relax the defect condition and simply require the row contraction to admit a cyclic vector. We show that cyclic pure row contractions can be "transformed" (in a precise technical sense) into compressions of the standard $d$-shift. Cyclic decompositions of the underlying Hilbert spaces are the natural tool to extend this fact to higher multipl…
$k$-Regular Factorizations and Joint Invariant Subspaces of Completely Non-Coisometric Row Contractions
研究完全非等距行压缩的特征函数的$k$-正则分解，建立联合不变子空间链与$k$-正则分解的一一对应，并给出函数模型和块上三角分解。
Typical properties of positive contractions and the invariant subspace problem
本文研究了典型正收缩在不同拓扑下的不变子空间性质，并证明了其在某些条件下不满足Abramovich, Aliprantis和Burkinshaw准则。
We answer affirmatively the problem left open in \cite{DSZ,GSZZ} and prove that for a finite Blaschke product $ϕ$, the minimal reducing subspaces of the Bergman space multiplier $M_ϕ$ are pairwise orthogonal and their number is equal to the number $q$ of connected components of the Riemann surface of $ϕ^{-1}\circ ϕ$. In particular, the double commutant $\{M_ϕ,M_ϕ^\ast\}'$ is abelian of dimension $q$. An analytic/arithmetic description of the minimal reducing subspaces of $M_ϕ$ is also provided, along with a list of all possible cases in degree of $ϕ$ equal to eight.
$k$-Regular Factorizations and Invariant Subspaces of Completely Non-Unitary Contractions
引入压缩算子的$k$-正则分解概念，建立其与完全非酉压缩的不变子空间链之间的一一对应，并构造显式泛函模型，同时引入交换$k$-压缩的对称$k$-正则元组，证明经典反例不满足该性质。
We provide a characterization of the commutant of analytic Toeplitz operators $T_B$ induced by finite Blachke products $B$ acting on weighted Bergman spaces which, as a particular instance, yields the case $B(z)=z^n$ on the Bergman space solved recently by by Abkar, Cao and Zhu. Moreover, it extends previous results by Cowen and Wahl in this context and applies to other Banach spaces of analytic functions such as Hardy spaces $H^p$ for $1<p<\infty$. Finally, we apply this approach to study the reducing subspaces of $T_{B}$ in weighted Bergman spaces.
This paper is devoted to the study of reducing subspaces for multiplication operator $M_ϕ$ on the Dirichlet space with symbol of finite Blaschke product. The reducing subspaces of $M_ϕ$ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surface to study the reducing subspaces of $M_ϕ$ on the Bergman space, and we discover a new way to study the Riemann surface for $ϕ^{-1}\circϕ$. By this means, we determine the reducing subspaces of $M_ϕ$ on the Dirichlet space when the order of $ϕ$ is $5$; $6$; $7$ and answer some questions of Douglas-Putin…
Using the tools of Sz.-Nagy--Foias theory of contractions, we describe in detail the invariant subspaces of the operator $ S\oplus S^* $, where $ S $ is the unilateral shift on a Hilbert space. This answers a question of Câmara and Ross.
The goal of the present paper is to push Sz.-Nagy--Foias model theory for a completely nonunitary Hilbert-space contraction operator $T$, to the case of a commuting pair of contraction operators $(T_1, T_2)$ having product $T = T_1 T_2$ which is completely nonunitary. The idea is to use the Sz.-Nagy-Foias functional model for $T$ as the model space also for the commutative tuple ($T_1, T_2)$ with $T = T_1 T_2$ equal to the usual Sz.-Nagy--Foias model operator, and identify what added structure is required to classify such commutative contractive factorizations $T = T_1 T_2$ up to unitary equiv…
In this paper, we provide the characterization of nonhomogeneous wavelet bi-frames. First of all we introduce the reducing subspaces of Sobolev spaces over local fields of prime characteristics and then characterize the nonhomogeneous wavelet bi frames over such fields.
We study reducing subspaces for an analytic multiplication operator M_{z^{n}} on the Bergman space L_{a}^{2}(A_{r}) of the annulus A_{r}, and we prove that M_{z^{n}} has exactly 2^n reducing subspaces. Furthermore, in contrast to what happens for the disk, the same is true for the Hardy space on the annulus. Finally, we extend the results to certain bilateral weighted shifts, and interpret the results in the context of complex geometry.
A particular case of results from [K2] is as follows. Let the unitary asymptote of a contraction $T$ contain the bilateral shift (of finite or infinite multiplicity). Then there exists an invariant subspace $\mathcal M$ of $T$ such that $T|_{\mathcal M}$ is similar to the unilateral shift of the same multiplicity. The proof is based on the Sz.-Nagy--Foias functional model for contractions. In the present paper this result is generalized to polynomially bounded operators, but in the simplest case. Namely, it is proved that if the unitary asymptote of a polynomially bounded operator $T$ contains…
Automorphic Nelson Dilations for Contractions and Invariant Subspace Tracking
研究\(n\times n\)严格收缩矩阵\(T\)的自守Nelson扩张，通过该扩张可快速获其最小等距和酉扩张，与经典理论相连。还证明每个\(T\)的Nelson扩张有特别性质，如强纠缠特征值函数或与\(z\)无关的约化子空间，同时研究了相关矩阵乘积特征值情况。
Regularity of the Sz.-Nagy and Foiaş Factorization of Characteristic Functions and Its Multivariable Analogue
研究具有不变子空间的压缩算子和行压缩算子特征函数分解的正则性，构造反例说明一般非正则，给出正则性刻画，确定几类正则算子，引入\(k -\)奇异分解概念并证明特定压缩算子特征函数不存在\(k -\)奇异分解。
In this paper, we mainly study geometric constructions of thin Blaschke products $B$ and reducing subspace problem of multiplication operators induced by such symbols $B$ on the Bergman space. Considering such multiplication operators $M_B$, we present a representation of those operators commuting with both $M_B$ and $M_B^*$. It is shown that for "most" thin Blaschke products $B$, $M_B$ is irreducible, i.e. $M_B$ has no nontrivial reducing subspace; and such a thin Blaschke product $B$ is constructed. As an application of the methods, it is proved that for "most" finite Blaschke products $ϕ$, …
For $b\in H^\infty_1$, the closed unit ball of $H^\infty$, the de Branges-Rovnyak spaces $\mathcal{H}(b)$ is a Hilbert space contractively contained in the Hardy space $H^2$ that is invariant by the backward shift operator $S^*$. We consider the reducing subspaces of the operator $S^{*2}|_{\mathcal{H}(b)}$. When $b$ is an inner function, $S^{*2}|_{\mathcal{H}(b)}$ is a truncated Toepltiz operator and its reducibility was characterized by Douglas and Foias using model theory. We use another approach to extend their result to the case where $b$ is extreme. We prove that if $b$ is extreme but not…
We prove that, in a large class of Banach lattices, the fixed space of each commuting family of positive linear contractions is a lattice subspace. As consequences, new cyclicity results for the peripheral point spectra of positive operators and semigroups are derived; we also pose an open problem that naturally occurs in this context. Finally, a variety of counterexamples is presented to point out some limits of our results.
Let $G$ be a finite pseudoreflection group, $Ω\subseteq \mathbb C^n$ be a bounded domain which is a $G$-space and $\mathcal H\subseteq\mathcal O(Ω)$ be an analytic Hilbert module possessing a $G$-invariant reproducing kernel. We study the structure of joint reducing subspaces of the multiplication operator $\mathbf M_{\boldsymbolθ}$ on $\mathcal H,$ where $\{θ_i\}_{i=1}^n$ is a homogeneous system of parameters associated to $G$ and $\boldsymbolθ= (θ_1, \ldots, θ_n)$ is a polynomial map of $\mathbb C^n$. We show that it admits a family $\{\mathbb P_\varrho\mathcal H:\varrho\in\widehat G\}$ of n…
On similarity to contractions of class $C_{\cdot 0}$ with finite defects
本文给出算子与具有有限亏格的 $C_{\cdot 0}$ 类压缩算子相似的判据及充分条件，基于移位型不变子空间和 Carleson--Newman 乘积。
In this paper, we study the reducing subspaces for the multiplication operator by a finite Blaschke product $ϕ$ on the Dirichlet space $D$. We prove that any two distinct nontrivial minimal reducing subspaces of $M_ϕ$ are orthogonal. When the order $n$ of $ϕ$ is $2$ or $3$, we show that $M_ϕ$ is reducible on $D$ if and only if $ϕ$ is equivalent to $z^n$. When the order of $ϕ$ is $4$, we determine the reducing subspaces for $M_ϕ$, and we see that in this case $M_ϕ$ can be reducible on $D$ when $ϕ$ is not equivalent to $z^4$. The same phenomenon happens when the order $n$ of $ϕ$ is not a prime n…
Quasianalytic contractions form the crucial class in the quest for proper invariant and hyperinvariant subspaces for asymptotically non-vanishing Hilbert space contractions. The property of quasianalycity relies on the concepts of unitary asymptote and $H^ı$-functional calculus. These objects can be naturally defined in the setting of polynomially bounded operators too, which makes possible to extend the study of quasianalycity from contractions to this larger class. Carrying out this program we pose also several interesting questions.
On unitary equivalence and reducing subspaces of analytic Toeplitz operator on vector-valued Hardy space
该研究证明向量值Hardy空间上的$T_{z^n}$算子酉等价于多份标量Hardy空间上$T_z$的直和，并结合矩阵与算子理论方法完整刻画了该算子的约化子空间。
The Stability and Density of Norm Attaining Operators on Reducing Subspaces
本文研究无限维希尔伯特空间上约化子空间范数达到算子的类$\beta(H)$，证明其在算子范数下稠密，并引入子类$\beta_0(H)$，刻画其有限秩与紧扰动稳定性，阐明谱结构。
The problem of block diagonalization for diagonally dominant symmetric block operator matrices with self-adjoint diagonal entries is considered. We show that a reasonable block diagonalization with respect to a reducing graph subspace requires a related skew-symmetric operator to be a strong solution to the associated Riccati equation. Under mild additional regularity conditions, we also establish that this skew-symmetric operator is a strong solution to the Riccati equation if and only if the graph subspace is reducing for the given operator matrix. These regularity conditions are shown to be…
Reducing and Invariant subspaces under two commuting shift operators
针对单位圆上取值于带重数 Hardy 空间的平方可积函数空间，刻画在双边移位与局部单边移位下的约化及不变子空间，条件推广了 Helson 与 Beurling-Lax-Halmos 定理。
Joint reducing subspaces and orthogonal decompositions of operators in an annulus
本文研究环带谱集对应的希尔伯特空间交换算子组，刻画$\mathbb A_r^n$-酉与等距算子结构，得到其Wold型分解并推广至压缩算子族。
Power partial isometries
本文完整刻画了幂部分等距算子完全非酉部分的约化、不变与超不变子空间，并特别给出了截断移位算子三类子空间的精确刻画。
Let $(S_1, S_2)$ be a bi-isometry, that is, a pair of commuting isometries $S_1$ and $S_2$ on a complex Hilbert space $\mathscr H.$ By the von Neumann-Wold decomposition, the hyper-range $\mathscr H_\infty(S_1):=\cap_{n=0}^\infty S^n_1\mathscr H$ of $S_1$ reduces $S_1$ to a unitary operator. Although $\mathscr H_\infty(S_1)$ is an invariant subspace for $S_2,$ in general, $\mathscr H_\infty(S_1)$ is not a reducing subspace for $S_2.$ We show that $\mathscr H_\infty(S_1)$ reduces $S_2$ to an isometry if and only if the subspaces $S_2(\ker S^*_1)$ and $\mathscr H_\infty(S_1)$ of $\mathscr H$ are…
Definition. Let J be a period-2 unitary operator (some people say J is reflection operator or reflection symmetry) and U be a linear operator. If U^*JU = J (resp. U^*JU >= J) then U is said to be J-isometry (resp. J-noncontraction). If U^*JU >= J and UJU^* >= J) then U is said to be J-binoncontraction). Theorem. If every J-isometry has nontrivial positive invariant subspace then every J-noncontraction has such a subspace. Theorem. If every J-binoncontractive J-isometry has maximal positive invariant subspace then every J-noncontraction has such a subspace. The article text is the complete text…
We review how some multianalytic inner functions of the Beurling type theorem are associated to row contractions following works of G.Popescu. Motivated by a result on weak Markov dilations, we define a notion of characteristic function for ergodic and coisometric row contractions with a one-dimensional invariant subspace for the adjoints. Our characteristic function is a complete unitary invariant for such tuples. Thereafter we extend the analysis of characteristic functions to contractive liftings of row contractions. We apply our theory to completely positive maps and explore applications t…
Given a self-adjoint involution J on a Hilbert space H, we consider a J-self-adjoint operator L=A+V on H where A is a possibly unbounded self-adjoint operator commuting with J and V a bounded J-self-adjoint operator anti-commuting with J. We establish optimal estimates on the position of the spectrum of L with respect to the spectrum of A and we obtain norm bounds on the operator angles between maximal uniformly definite reducing subspaces of the unperturbed operator A and the perturbed operator L. All the bounds are given in terms of the norm of V and the distances between pairs of disjoint s…
Minimum attaining operators on reducing subspaces: Spectral structure and density
本文引入可分希尔伯特空间上最小达到算子的新子类$\boldsymbol{\textit{M}}_r(H)$，刻画其算子结构与谱性质，证明该类在算子范数下于$\boldsymbol{\textit{B}}(H)$中稠密，还得到其正规算子表示定理。
Canonical decomposition of dissipative linear relations
本文利用 Sz. Nagy-Foiaş-Langer 与 von Neumann-Wold 分解，通过 Z 变换构造耗散线性关系的两种典范分解，实现自伴部分与完全非自伴部分的分离。
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