Backward Shift Invariant Subspaces Research Articles

De Branges-Rovnyak spaces generated by row Schur functions with mate

In this paper, we study the de Branges-Rovnyak spaces H(B) generated by row Schur functions B with mate a. We prove that the polynomials are dense in H(B), and characterize the backward shift invariant subspaces of H(B). We then describe the cyclic vectors in H(B) when B is of finite rank and dim⁡(aH2)⊥<∞.

Study of nearly invariant subspaces with finite defect in Hilbert spaces

In this article, we briefly describe nearly $$T^{-1}$$ invariant subspaces with finite defect for a shift operator T having finite multiplicity acting on a separable Hilbert space $${\mathcal {H}}$$ as a generalization of nearly $$T^{-1}$$ invariant subspaces introduced by Liang and Partington in Complex Anal. Oper. Theory 15(1) (2021) 17 pp. In other words, we characterize nearly $$T^{-1}$$ invariant subspaces with finite defect in terms of backward shift invariant subspaces in vector-valued Hardy spaces by using Theorem 3.5 in Int. Equations Oper. Theory 92 (2020) 1–15. Furthermore, we also provide a concrete representation of the nearly $$T_B^{-1}$$ invariant subspaces with finite defect in a scale of Dirichlet-type spaces $${\mathcal {D}}_\alpha $$ for $$\alpha \in [-1,1]$$ corresponding to any finite Blashcke product B, as was done recently by Liang and Partington for defect zero case (see Section 3 of Complex Anal. Oper. Theory 15(1) (2021) 17 pp).

Compressions of kth-order slant Toeplitz operators to model spaces

In this paper, we consider compressions of kth-order slant Toeplitz operators to the backward shift-invariant subspaces of the classical Hardy space H2. In particular, we characterize these operators using compressed shifts and finite-rank operators of special kind.

Kernels of perturbed Toeplitz operators in vector-valued Hardy spaces

Recently, Liang and Partington (Integr Equ Oper Theory 92(4): 35, 2020) show that kernels of finite-rank perturbations of Toeplitz operators are nearly invariant with finite defect under the backward shift operator acting on the scalar-valued Hardy space. In this article, we provide a vectorial generalization of a result of Liang and Partington. As an immediate application, we identify the kernel of perturbed Toeplitz operator in terms of backward shift-invariant subspaces in various important cases by applying the recent theorem (see Chattopadhyay et al. in Integr Equ Oper Theory 92(6): 52, 2020, Theorem 3.5 and O’Loughlin in Complex Anal Oper Theory 14(8): 86, 2020, Theorem 3.4) in connection with nearly invariant subspaces of finite defect for the backward shift operator acting on the vector-valued Hardy space.

Idempotent, model, and Toeplitz operators attaining their norms

We study idempotent, model, and Toeplitz operators that attain the norm. Notably, we prove that if Q is a backward shift invariant subspace of the Hardy space H2(D), then the model operator SQ attains its norm. Here SQ=PQMz|Q, the compression of the shift Mz on the Hardy space H2(D) to Q.

Backward Shift and Nearly Invariant Subspaces of Fock-type Spaces

Abstract We study the structure of the backward shift invariant and nearly invariant subspaces in weighted Fock-type spaces ${\mathcal{F}}_W^p$, whose weight is not necessarily radial. We show that in the spaces ${\mathcal{F}}_W^p$, which contain the polynomials as a dense subspace (in particular, in the radial case), all nontrivial backward shift invariant subspaces are of the form $\mathcal{P}_n$, that is, finite-dimensional subspaces consisting of polynomials of degree at most $n$. In general, the structure of the nearly invariant subspaces is more complicated. In the case of spaces of slow growth (up to zero exponential type), we establish an analogue of de Branges’ ordering theorem. We then construct examples that show that the result fails for general Fock-type spaces of larger growth.

Multi-orbital Frames Through Model Spaces

We characterize the normal operators A on \(\ell ^2\) and the elements \(a^i \in \ell ^2\), with \(1\le i\le m\), such that the sequence $$\begin{aligned} \{ A^n a^1, \ldots , A^n a^m \}_{n\ge 0} \end{aligned}$$is a frame. The characterization makes strong use of the pseudo-hyperbolic metric of \( {{\mathbb {D}}} \) and is given in terms of the backward shift invariant subspaces of \(H^2( {{\mathbb {D}}} )\) associated to finite products of interpolating Blaschke products.

The Beurling-Lax-Halmos theorem for infinite multiplicity

In this paper we consider several questions emerging from the Beurling-Lax-Halmos Theorem, which characterizes the shift-invariant subspaces of vector-valued Hardy spaces. The Beurling-Lax-Halmos Theorem states that a backward shift-invariant subspace is a model space H(Δ)≡HE2⊖ΔHE2, for some inner function Δ. Our first question calls for a description of the set F in HE2 such that H(Δ)=EF⁎, where EF⁎ denotes the smallest backward shift-invariant subspace containing the set F. In our pursuit of a general solution to this question, we are naturally led to take into account a canonical decomposition of operator-valued strong L2-functions. Next, we ask: Is every shift-invariant subspace the kernel of a (possibly unbounded) Hankel operator? Consideration of the question on the structure of shift-invariant subspaces leads us to study and coin a new notion of “Beurling degree” for an inner function. We then establish a deep connection between the spectral multiplicity of the model operator (the truncated backward shift) and the Beurling degree of the corresponding characteristic function. At the same time, we consider the notion of meromorphic pseudo-continuations of bounded type for operator-valued functions, and then use this notion to study the spectral multiplicity of model operators between separable complex Hilbert spaces. In particular, we consider the case of multiplicity-free: more precisely, for which characteristic function Δ of the model operator T does it follow that T is multiplicity-free, i.e., T has multiplicity 1? We show that if Δ has a meromorphic pseudo-continuation of bounded type in the complement of the closed unit disk and the adjoint of the flip of Δ is an outer function, then T is multiplicity-free.

Representing Kernels of Perturbations of Toeplitz Operators by Backward Shift-Invariant Subspaces

It is well known that the kernel of a Toeplitz operator is nearly invariant under the backward shift $S^*$. This paper shows that kernels of finite-rank perturbations of Toeplitz operators are nearly $S^*$-invariant with finite defect. This enables us to apply a recent theorem by Chalendar--Gallardo--Partington to represent the kernel in terms of backward shift-invariant subspaces, which we identify in several important cases.

Backward shift invariant subspaces in reproducing kernel Hilbert spaces

In this note, we describe the backward shift invariant subspaces for an abstract class of reproducing kernel Hilbert spaces. Our main result is inspired by a result of Sarason concerning de Branges-Rovnyak spaces (the non-extreme case). Furthermore, we give new applications in the context of the range space of co-analytic Toeplitz operators and sub-Bergman spaces.

Independence of Vector-Valued Functions Associated with Kernels of Block Hankel Operators

For a block Hankel operator $$H_\Phi $$ with a matrix function symbol $$\Phi \in L^2_{M_{n\times m}}$$ , it is well-known that $$\ker H_\Phi = \Theta H^2_{\mathbb {C}^r}$$ for a natural number r and an $$m\times r$$ matrix inner function $$\Theta $$ if $$\ker H_\Phi $$ is nonempty. It will be shown that the size of the matrix inner function $$\Theta $$ associated with $$\ker H_\Phi $$ is closely related with a certain degree of independence of the column vectors of $$\Phi $$ , which will be defined in this paper. As an important application, the shape of shift invariant, or, backward shift invariant subspaces of $$H^2_{\mathbb {C}^n}$$ generated by finite elements will be studied.

Toeplitz operators on backward shift invariant subspaces of Hp

We extend results on compressed Toeplitz operators on the backward shift invariant subspaces of H2 to the context of the spaces Hp, 1<p<∞.

Range Spaces of Co-Analytic Toeplitz Operators

AbstractIn this paper we discuss the range of a co-analytic Toeplitz operator. These range spaces are closely related to de Branges–Rovnyak spaces (in some cases they are equal as sets). In order to understand its structure, we explore when the range space decomposes into the range of an associated analytic Toeplitz operator and an identifiable orthogonal complement. For certain cases, we compute this orthogonal complement in terms of the kernel of a certain Toeplitz operator on the Hardy space, where we focus on when this kernel is a model space (backward shift invariant subspace). In the spirit of Ahern–Clark, we also discuss the non-tangential boundary behavior in these range spaces. These results give us further insight into the description of the range of a co-analytic Toeplitz operator as well as its orthogonal decomposition. Our Ahern–Clark type results, which are stated in a general abstract setting, will also have applications to related sub-Hardy Hilbert spaces of analytic functions such as the de Branges–Rovnyak spaces and the harmonically weighted Dirichlet spaces.

Asymmetric Truncated Toeplitz Operators of Rank One

A truncated Toeplitz operator is a compression of the multiplicationoperator to a backward shift invariant subspace of the Hardy space H^2. Anasymmetric truncated Toeplitz operator is a compression of the multiplication operator that acts between two different backward shift invariant subspaces of H^2. All rank-one truncated Toeplitz operators have been described by Sarason. Here, we characterize all rank-one asymmetric truncated Toeplitz operators. This completes the description given by Łanucha for asymmetric truncated Toeplitz operators on finite-dimensional backward shift invariant subspaces.

Adjoint of the Toeplitz operator with the singular inner function

Let ψδ1 be the singular inner function associated with the unit point mass at z=1. We shall give an infinite dimensional closed subspace M of kerTψδ1⁎, the kernel of the adjoint of the Toeplitz operator Tψδ1, satisfying that for every nonzero v in M, the smallest backward shift invariant subspace of H2 containing v coincides with kerTψδ1⁎.

Kernels of Toeplitz operators on the Hardy space over the bidisk

We study the kernels of Toeplitz operators on the Hardy space on the bidisk. We first give a sufficient condition for a general symbol to be antiholomorphic under the assumption that the kernel of the corresponding Toeplitz operator contains a backward shift invariant subspace. As an application, we construct an invariant subspace whose its orthogonal complement can not be the kernel of any Toeplitz operator. Also, we give a characterization on a Toeplitz operator for which the orthogonal complement of its kernel is generated by certain inner functions. Finally, we describe all backward shift invariant subspaces which are in the kernels of Toeplitz operators with homogeneous type symbols. As an application, we show that there is a Toeplitz operator for which its kernel has dimension of any given integer and the orthogonal complement of the kernel is generated by two functions. Our result shows that there are higher dimensional phenomena.

Backward shift invariant subspaces with applications to band preserving and phase retrieval problems

The band preserving and phase retrieval problems have long been interested and studied. In this paper, we, for the first time, give solutions to these problems in terms of backward shift invariant subspaces. The backward shift method among other methods seems to be direct and natural. We show that a function , with , that makes the band of fg to be within that of f if and only if g divided by an inner function related to f, belongs to some backward shift invariant subspace in relation to f. By the construction of backward shift invariant space, the solution g is further explicitly represented through the span of the rational function system whose zeros are those of the Laplace transform of f. As an application, we also use the backward shift method to give a characterization for the solutions of the phase retrieval problem. Copyright © 2015 John Wiley & Sons, Ltd.

New aspects of Beurling–Lax shift invariant subspaces

In terms of forward and backward shift invariant subspaces, we characterize functions in Hardy spaces, or, analytic signals in the terminology of signal analysis, through multiplications between analytic and conjugate analytic signals. As applications, we give some necessary and sufficient conditions for solutions of the Bedrosian equation H(fg)=f(Hg) when f or g is a bandlimited signal. We also solve the band preserving problem by means of the shift invariant subspace method, which establishes some necessary and sufficient conditions on the functions f that make fg have bandwidth within that of the function g.

Spectral models for orthonormal wavelets and related algorithms

This work presents new methods for constructing orthonormal wavelets from certain families of Hardy functions. Inner functions and the corresponding backward shift invariant subspaces are in the core of the structure of these families. The new algorithms focus on the local shape of the wavelets, a property making them especially useful for pattern recognition.

Boundary values in range spaces of co-analytic truncated Toeplitz operators

Functions in backward shift invariant subspaces have nice analytic continuation properties outside the spectrum of the inner function de ning the space. Inside the spectrum of the inner function, Ahern and Clark showed that under some distribution condition on the zeros and the singular measure of the inner function, it is possible to obtain non-tangential boundary values of every function in the backward shift invariant subspace as well as for their derivatives up to a certain order. Here we will investigate, at least when the inner function is a Blaschke product, the non- tangential boundary values of the functions of the backward shift invariant subspace after having applied a co-analytic (truncated) Toeplitz operator. There appears to be a smoothing effect.