Shift-invariant Subspaces Research Articles

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.

On Generalizations of Sampling Theorem and Stability Theorem in Shift-Invariant Subspaces of Lebesgue and Wiener Amalgam Spaces with Mixed-Norms

In this paper, we establish generalized sampling theorems, generalized stability theorems and new inequalities in the setting of shift-invariant subspaces of Lebesgue and Wiener amalgam spaces with mixed-norms. A convergence theorem of general iteration algorithms for sampling in some shift-invariant subspaces of Lp→(Rd) are also given.

Harmonic maps and shift-invariant subspaces

With the help of operator-theoretic methods, we derive new and powerful criteria for finiteness of the uniton number for a harmonic map from a Riemann surface to the unitary group $${{\,\mathrm{U}\,}}(n)$$ . These use the Grassmannian model where harmonic maps are represented by families of shift-invariant subspaces of $$L^2(S^1,{{\mathbb {C}}}^n)$$ ; we give a new description of that model.

New Generalizations and Results in Shift-Invariant Subspaces of Mixed-Norm Lebesgue Spaces \({L_{\vec{p}}(\mathbb{R}^d)}\)

In this paper, we establish new generalizations and results in shift-invariant subspaces of mixed-norm Lebesgue spaces Lp→(Rd). We obtain a mixed-norm Hölder inequality, a mixed-norm Minkowski inequality, a mixed-norm convolution inequality, a convolution-Hölder type inequality and a stability theorem to mixed-norm case in the setting of shift-invariant subspace of Lp→(Rd). Our new results unify and refine the existing results in the literature.

Symmetric shift-invariant subspaces and harmonic maps

The Grassmannian model represents harmonic maps from Riemann surfaces by families of shift-invariant subspaces of a Hilbert space. We impose a natural symmetry condition on the shift-invariant subspaces that corresponds to considering an important class of harmonic maps into symmetric and k-symmetric spaces. Using an appropriate description of such symmetric shift-invariant subspaces we obtain new results for the corresponding extended solutions, including how to obtain primitive harmonic maps from certain harmonic maps into the unitary group.

The Structure of Finitely Generated Shift-Invariant Subspaces on Locally Compact Abelian Groups

In this paper, we characterize finitely generated shift-invariant subspaces of $$L^2(G)$$ , where G is a locally compact abelian group. In particular, we give a formula for the coefficients in the known representation of the Fourier transform of the elements of finitely generated shift-invariant subspaces. Also, certain orthogonalization procedure for generators which is reminiscent of the Gram–Schmidt orthogonalization process is given.

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.

Multiparameter Eigenvalue Problems and Shift-invariance

We discuss four eigenvalue problems of increasing generality and complexity: rooting a univariate polynomial, solving the polynomial eigenvalue problem, rooting a set of multivariate polynomials and solving multi-parameter eigenvalue problems. In doing so, we provide a unifying framework for solving these eigenvalue problems, where we exploit properties of (block-) (multi-) shift-invariant subspaces and use multi-dimensional realization algorithms.

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.

Nearly Invariant Subspaces with Applications to Truncated Toeplitz Operators

In this paper we first study the structure of the scalar and vector-valued nearly invariant subspaces with a finite defect. We then subsequently produce some fruitful applications of our new results. We produce a decomposition theorem for the vector-valued nearly invariant subspaces with a finite defect. More specifically, we show every vector-valued nearly invariant subspace with a finite defect can be written as the isometric image of a backwards shift invariant subspace. We also show that there is a link between the vector-valued nearly invariant subspaces and the scalar-valued nearly invariant subspaces with a finite defect. This is a powerful result which allows us to gain insight in to the structure of scalar subspaces of the Hardy space using vector-valued Hardy space techniques. These results have far reaching applications, in particular they allow us to develop an all encompassing approach to the study of the kernels of: the Toeplitz operator, the truncated Toeplitz operator, the truncated Toeplitz operator on the multiband space and the dual truncated Toeplitz operator.

Random sampling in multiply generated shift-invariant subspaces of mixed Lebesgue spaces [formula omitted

We mainly study the random sampling and reconstruction in multiply generated shift-invariant subspaces Vp,q(Φ) of mixed Lebesgue spaces Lp,q(R×Rd). Under suitable conditions for the generators Φ, we can prove that if the sampling sizes are large enough for both variables, the sampling stability holds with high probability for all functions in Vp,q(Φ) whose energy is concentrated on a compact subset. Finally, a reconstruction algorithm based on random samples is given for functions in a finite dimensional subspace.

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.

Semi-Average Sampling for Shift-Invariant Signals in a Mixed Lebesgue Space

This article mainly studies the nonuniform and semi-average sampling of time-varying shift-invariant signals living in a mixed Lebesgue space under the condition that the generator of the shift-invariant subspace belongs to a hybrid-norm space of mixed form depending on the orders p and q. First, the sampling stability for two kinds of semi-average sampling functionals is established. Then, we give the corresponding iterative approximation projection algorithms with exponential convergence for recovering the time-based shift-invariant signals from the semi-average samples.

Positive Weight Function and Classification of g-Frames

Given a positive weight function and an isometry map on a Hilbert spaces we study a class of linear maps which is a g-frame, g-Riesz basis, and a g-orthonormal basis for with respect to in terms of the weight function. We apply our results to study the frame for shift-invariant subspaces on the Heisenberg group.

Convolution Systems on Discrete Abelian Groups as a Unifying Strategy in Sampling Theory

A regular sampling theory in a multiply generated unitary invariant subspace of a separable Hilbert space $${\mathcal {H}}$$ is proposed. This subspace is associated to a unitary representation of a countable discrete abelian group G on $${\mathcal {H}}$$. The samples are defined by means of a filtering process which generalizes the usual sampling settings. The multiply generated setting allows to consider some examples where the group G is non-abelian as, for instance, crystallographic groups. Finally, it is worth to mention that classical average or pointwise sampling in shift-invariant subspaces are particular examples included in the followed approach.

Nonuniform average sampling in multiply generated shift-invariant subspaces of mixed Lebesgue spaces

In this paper, we study nonuniform average sampling problem in multiply generated shift-invariant subspaces of mixed Lebesgue spaces. We discuss two types of average sampled values: average sampled values [Formula: see text] generated by single averaging function and average sampled values [Formula: see text] generated by multiple averaging functions. Two fast reconstruction algorithms for these two types of average sampled values are provided.

Multivariate vector sampling expansion in shift-invariant subspaces

Sampling theorems on a shift-invariant subspace are having a significant impact, since they avoid most of the problems associated with classical Shannon's theory. Vector sampling theorems on a shift-invariant subspace which are motivated by applications in multi-channel deconvolution and multi-source separation are active field of study. In this paper, we consider vector sampling theorems on a multivariate vector shift-invariant subspace. We give a multivariate vector sampling expansion on a multivariate vector shift-invariant subspace. Some equivalence conditions for the multivariate vector sampling expansion to hold are given. We also give several examples to illustrate the main result.

Multivariate vector sampling expansion in shift-invariant subspaces

Multivariate vector sampling expansion in shift-invariant subspaces