Reducing Subspaces Research Articles

A geometric approach to the compressed shift operator on the Hardy space over the bidisk

This paper studies the compressed shift operator Sz on the Hardy space over the bidisk via the geometric approach. We calculate the spectrum and essential spectrum of Sz on the Beurling type quotient modules induced by rational inner functions, and give a complete characterization for Sz⁎ to be a Cowen-Douglas operator. Then we extend the concept of Cowen-Douglas operator to be the generalized Cowen-Douglas operator, and show that Sz⁎ is a generalized Cowen-Douglas operator. Moreover, we establish the connection between the reducibility of the Hermitian holomorphic vector bundle induced by kernel spaces and the reducibility of the generalized Cowen-Douglas operator. By using the geometric approach, we study the reducing subspaces of Sz on certain polynomial quotient modules.

Reducibility of Self-Adjoint Linear Relations and Application to Generalized Nevanlinna Functions

Necessary and sufficient conditions for reducidibility of a self-adjoint linear relation in a Krein space are given. Then a generalized Nevanlinna function $Q$, represented by a self-adjoint linear relation $A$, is decomposed by means of the reducing subspaces of $A$. The sum of two functions $Q_{i}{\in N}_{\kappa_{i}}\left( \mathcal{H} \right),\thinspace i=1,\thinspace 2$, minimally represented by the triplets $\left( \mathcal{K}_{i},A_{i},\Gamma_{i} \right)$, is also studied. For that purpose, a model $( \tilde{\mathcal{K}},\tilde{A},\tilde{\Gamma } )$ to represent $Q:=Q_{1}+Q_{2}$ in terms of $\left( \mathcal{K}_{i},A_{i},\Gamma_{i} \right)$ is created. By means of that model, necessary and sufficient conditions for $\kappa =\kappa_{1}+\kappa_{2}$ are proven in analytic terms. At the end, it is explained how degenerate Jordan chains of the representing relation $A$ affect reducing subspaces of $A$ and decomposition of the corresponding function $Q$.

Reducing subspaces for rank-one perturbations of normal operators

We study the existence of reducing subspaces for rank-one perturbations of diagonal operators and, in general, of normal operators of uniform multiplicity one. As we will show, the spectral picture will play a significant role in order to prove the existence of reducing subspaces for rank-one perturbations of diagonal operators whenever they are not normal. In this regard, the most extreme case is provided when the spectrum of the rank-one perturbation of a diagonal operator $T=D + u\otimes v$ (uniquely determined by such expression) is contained in a line, since in such a case $T$ has a reducing subspace if and only if $T$ is normal. Nevertheless, we will show that it is possible to exhibit non-normal operators $T=D + u\otimes v$ with spectrum contained in a circle either having or lacking non-trivial reducing subspaces. Moreover, as far as the spectrum of $T$ is contained in any compact subset of the complex plane, we provide a characterization of the reducing subspaces $M$ of $T$ such that the restriction $T\mid _M$ is normal. In particular, such characterization allows us to exhibit rank-one perturbations of completely normal diagonal operators (in the sense of Wermer) lacking reducing subspaces. Furthermore, it determines completely the decomposition of the underlying Hilbert space in an orthogonal sum of reducing subspaces in the context of a classical theorem due to Behncke on essentially normal operators.

Common reducing subspaces and decompositions of contractions

Abstract A commuting triple of Hilbert space operators ( A , B , P ) {(A,B,P)} , for which the closed tetrablock 𝔼 ¯ {\overline{\mathbb{E}}} is a spectral set, is called a tetrablock-contraction or simply an 𝔼 {\mathbb{E}} -contraction, where 𝔼 = { ( x 1 , x 2 , x 3 ) ∈ ℂ 3 : 1 - x 1 ⁢ z - x 2 ⁢ w + x 3 ⁢ z ⁢ w ≠ 0 ⁢ whenever ⁢ | z | ≤ 1 ⁢ and ⁢ | w | ≤ 1 } ⊂ ℂ 3 , \mathbb{E}=\bigl{\{}(x_{1},x_{2},x_{3})\in\mathbb{C}^{3}:1-x_{1}z-x_{2}w+x_{3}% zw\neq 0\text{ whenever }\lvert z\rvert\leq 1\text{ and }\lvert w\rvert\leq 1% \bigr{\}}\subset\mathbb{C}^{3}, is a polynomially convex domain which is naturally associated with the μ-synthesis problem. By applications of the theory of 𝔼 {\mathbb{E}} -contractions, we obtain the following main results in this article: (i) It is well known that any number of doubly commuting isometries admit a Wold-type decomposition. Also, examples from the literature show that such decomposition does not hold in general for commuting isometries. We generalize this result to any number of doubly commuting contractions. Indeed, we present a canonical decomposition to any family of doubly commuting contractions 𝒰 = { T λ : λ ∈ Λ } {\mathcal{U}=\{T_{\lambda}:\lambda\in\Lambda\}} acting on a Hilbert space ℋ {\mathcal{H}} and show that ℋ {\mathcal{H}} continues to decompose orthogonally into joint reducing subspaces of 𝒰 {\mathcal{U}} until all members of 𝒰 {\mathcal{U}} split together into unitaries and completely non-unitary (c.n.u.) contractions. Naturally, as a special case we achieve an analogous Wold decomposition for any number of doubly commuting isometries. (ii) Further, we show that a similar canonical decomposition is possible for any family of doubly commuting c.n.u. contractions, where all members jointly orthogonally decompose into pure isometries (i.e. unilateral shifts) and completely non-isometry (c.n.i.) contractions. (iii) We give an alternative proof to the canonical decomposition of an 𝔼 {\mathbb{E}} -contraction and apply that to establish independently the following result due to Eschmeier: for any finitely many commuting contractions T 1 , … , T n {T_{1},\dots,T_{n}} acting on a Hilbert space ℋ {\mathcal{H}} , the space ℋ {\mathcal{H}} admits an orthogonal decomposition ℋ = ℋ 1 ⊕ ℋ 2 {\mathcal{H}=\mathcal{H}_{1}\oplus\mathcal{H}_{2}} , where ℋ 1 {\mathcal{H}_{1}} is the maximal common reducing subspace for T 1 , … , T n {T_{1},\dots,T_{n}} such that T 1 | ℋ 1 , … , T n | ℋ 1 {T_{1}\lvert_{\mathcal{H}_{1}},\dots,T_{n}\lvert_{\mathcal{H}_{1}}} are unitaries.

Minimal Reducing Subspaces of Weighted Pseudo Shifts

Minimal Reducing Subspaces of Weighted Pseudo Shifts

A characterization of nonhomogeneous wavelet bi-frames for reducing subspaces of Sobolev spaces

For nonhomogeneous wavelet bi-frames in a pair of dual spaces (H^{s}(mathbb{R}^{d}), H^{-s}(mathbb{R}^{d})) with sneq 0, smoothness and vanishing moment requirements are separated from each other, that is, one system is for smoothness and the other one for vanishing moments. This gives us more flexibility to construct nonhomogeneous wavelet bi-frames than in L^{2}(mathbb{R}^{d}). In this paper, we introduce the reducing subspaces of Sobolev spaces, and characterize the nonhomogeneous wavelet bi-frames under the setting of a general pair of dual reducing subspaces of Sobolev spaces.

Reducibility for a Class of Analytic Multipliers on Sobolev Disk Algebra

We prove the reducibility of analytic multipliers Mϕ with a class of finite Blaschke products symbol ϕ on the Sobolev disk algebra R(ⅅ). We also describe their nontrivial minimal reducing subspaces.

Reducing Subspaces of Complex Symmetric Operators

An operator T on a separable, infinite dimensional, complex Hilbert space $${\mathcal {H}}$$ is called complex symmetric if T has a symmetric matrix representation relative to some orthonormal basis for $${\mathcal {H}}$$ . This paper aims to describe reducing subspaces of complex symmetric operators from the view point of approximation. In particular, given a complex symmetric operator T, $$1\le n\le \aleph _0$$ and $$\varepsilon >0$$ , it is proved that there exists a compact operator K with $$\Vert K\Vert <\varepsilon $$ such that $$T+K$$ is complex symmetric and has exactly n minimal reducing subspaces.

Reducing and minimal reducing subspaces of slant Toeplitz operators

In this paper we determine all the minimal reducing subspaces for the slant Toeplitz operator $$A_{\varphi }$$ where $$\varphi (z)=z^N$$ for any integer N. We show that $$A_{\varphi }$$ has infinitely many minimal reducing subspaces in $$L^2({\mathbf {T}})$$. For every minimal reducing subspace M there exists an N-transparent function $$f\in L^2({\mathbb {T}})$$ such that $$M=X_f$$ where f is unique upto a scalar multiple.

Brownian Type Parts of Operators in Hilbert Spaces

We obtain some reducing (or just invariant) subspaces for a Hilbert space operator on which it acts as a Brownian type 2-isometry. More exactly Brownian isometric (unitary) parts as well as more general quasi-Brownian isometric (unitary) parts of an operator are investigated. They are explicitely described for a 2-isometry.

Reducing Subspaces of Analytic Toeplitz Operators on the Bergman Space of the Annulus

Let $$\mathbb {A}_{r}$$ be the annulus $$\{z\mid \,|z|<r<1\}$$ in the complex plane, $$L_{a}^{2}(\mathbb {A}_{r})$$ be the Bergman space on $$\mathbb {A}_{r}$$ , B be a finite Blaschke product $$B(z)=e^{i\theta }\prod \nolimits _{i=1}^{N}\frac{z-\alpha _{i}}{1-\overline{\alpha _{i}}z}$$ with $$|\alpha _{i}|<r$$ for $$1\le i\le N$$ . In this case, local inverses of B on $$\mathbb {A}_{r}$$ consist of a cyclic group with order N. It is shown that there is an one-to-one correspondence between a minimal reducing subspace of the Toeplitz operator $$T_{B}$$ on $$L_{a}^{2}(\mathbb {A}_{r})$$ and a character of the cyclic group, reducing subspaces of Toeplitz operators are studied from an algebraic point of view and Douglas and Kim’s result is generalized.

The reducibility of compressed shifts on Beurling type quotient modules over the bidisk

In this paper, we study the compressed shift operator Sz1 on the Beurling-type quotient module Kθ of Hardy space H2(D2) over the bidisk. Firstly, we give a necessary and sufficient condition such that Sz1 has nontrivial pure isometry reducing subspace. As an application, we show that Sz1 has Agler reducing subspaces if and only if θ is the product of two one variable inner functions. Secondly, for a rational inner function with degree (n,1), we show that Sz1 is reducible on Kθ if and only if Sz1 has Agler reducing subspaces. Furthermore, we study the case when the rational inner functions have degree (n,2), and this case is quite different from that the degree of θ is (n,1).

Common reducing subspaces of several weighted shifts with operator weights

We characterize common reducing subspaces of several weighted shifts with operator weights. As applications, we study the common reducing subspaces of the multiplication operators by powers of coordinate functions on Hilbert spaces of holomorphic functions in several variables. The identification of reducing subspaces also leads to structure theorems for the commutants of von Neumann algebras generated by these multiplication operators. This general approach applies to weighted Hardy spaces, weighted Bergman spaces, Drury–Arveson spaces and Dirichlet spaces of the unit ball or polydisk uniformly.

Minimal reducing subspaces of an operator-weighted shift

We introduce a family T consisting of invertible matrices with exactly one nonzero entry in each row and each column. The elements of T are possibly mutually noncommuting, and they need not be normal or self-adjoint. We consider an operator-valued unilateral weighted shift W with a uniformly bounded sequence of weights belonging to T, and we describe its minimal reducing subspaces.

Matrix convex sets without absolute extreme points

This article shows the existence of a class of closed bounded matrix convex sets which do not have absolute extreme points. The sets we consider are noncommutative sets, KX, formed by taking matrix convex combinations of a single tuple X. In the case that X is a tuple of compact operators with no nontrivial finite dimensional reducing subspaces, KX is a closed bounded matrix convex set with no absolute extreme points.A central goal in the theory of matrix convexity is to find a natural notion of an extreme point in the dimension free setting which is minimal with respect to spanning. Matrix extreme points are the strongest type of extreme point known to span matrix convex sets; however, they are not necessarily the smallest set which does so. Absolute extreme points, a more restricted type of extreme points that are closely related to Arveson's boundary, enjoy a strong notion of minimality should they span. This result shows that matrix convex sets may fail to be spanned by their absolute extreme points.

Reducing subspaces of multiplication operators on weighted Hardy spaces over bidisk

We consider weighted Hardy spaces over bidisk ${\mathbb D}^2$ which generalize the weighted Bergman spaces $A_\alpha^2({\mathbb D}^2)$. Let $z,w$ be coordinate functions and $M_{z^Nw^N}$ the multiplication by $z^Nw^N$ for a natural number $N$. In this paper, we study the reducing subspaces of $M_{z^Nw^N}$. In particular, we obtain the minimal reducing subspaces of $M_{zw}$.

Reducing subspaces for multiplication operators on the Dirichlet space through local inverses and Riemann surfaces

Abstract This paper gives a full characterization of the reducing subspaces for the multiplication operator Mϕ on the Dirichlet space with symbol of finite Blaschke product ϕ of order 5I 6I 7. The reducing subspaces of Mϕ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surfaces to study the reducing subspaces of Mϕ on the Bergman space. By this means, we determine the reducing subspaces of Mϕ on the Dirichlet space and answer some questions of Douglas-Putinar-Wang in [6].

Restriction of Toeplitz Operators on Their Reducing Subspaces

We study the restrictions of analytic Toeplitz operator on its minimal reducing subspaces for the unit disc and construct their models on slit domains. Furthermore, it is shown that Tzn is similar to the sum of n copies of the Bergman shift.

A Note on Reducing Subspaces of Toeplitz Operator on the Weighted Analytic Function Spaces of the Bidisk Hw2D2

We discussed the reducing subspaces of Toeplitz operators Tz1Nz2M N≠M on the weighted analytic function spaces of bidisk Hw2(D2). The result shows that if the weight w is of type-I, the structure of reducing subspaces of Tz1Nz2M N≠M on Hw2D2 is very simple.

Weak Nonhomogeneous Wavelet Bi-Frames for Reducing Subspaces of Sobolev Spaces

ABSTRACTAs homogeneous wavelet bi-frames derived from a pair of refinable functions, nonhomogeneous wavelet bi-frames admit fast wavelet transform, which is one of main concerns in wavelet analysis. Simultaneously, for a nonhomogeneous wavelet bi-frame in a pair of dual spaces with s≠0, smoothness and vanishing moment requirements are separated from each other, that is, one system is for smoothness and the other for vanishing moments. This gives us more flexibility to construct nonhomogeneous wavelet bi-frames than in . However, the requirement of the Bessel property of affine systems is always needed. For refinable-function based wavelet bi-frames, such a requirement is imposed on refinable functions in different ways. This is not what we expect. So a natural question involves what we are expecting from a pair of general refinable functions. For this purpose, we introduce the notion of the reducing subspace of a Sobolev space and the notion of the weak nonhomogeneous wavelet bi-frame in a general pair of dual reducing subspaces of . We characterize reducing subspaces of a Sobolev space. Then, under the setting of reducing subspaces, we study the properties of weak nonhomogeneous wavelet bi-frames, obtain a construction of weak nonhomogeneous wavelet bi-frames from a pair of refinable functions, and derive a fast wavelet algorithm associated with such weak nonhomogeneous wavelet bi-frames.