Characterization of the closedness of the sum of two shift-invariant spaces

We first present a formula for the supremum cosine angle between two closed subspaces of a separable Hilbert space under the assumption that the ‘generators’ form frames for the subspaces. We then characterize the conditions that the sum of two, not necessarily finitely generated, shift-invariant subspaces of L²(Rd) be closed. If the fibers of the generating sets of the shift-invariant subspaces form frames for the fiber spaces a.e., which is satisfied if the shift-invariant subspaces are finitely generated or if the shifts of the generating sets form frames for the respective subspaces, then the characterization is given in terms of the norms of possibly infinite matrices. In particular, if the shift-invariant subspaces are finitely generated, then the characterization is given wholly in terms of the norms of finite matrices.     

A unified characterization of reproducing systems generated by a finite family

This article presents a general result from the study of shift-invariant spaces that characterizes tight frame and dual frame generators for shift-invariant subspaces of L²(ℝⁿ). A number of applications of this general result are then obtained, among which are the characterization of tight frames and dual frames for Gabor and wavelet systems.     

Order of linear approximation from shift-invariant spaces

A Fourier analysis approach is taken to investigate the approximation order of scaled versions of certain linear operators into shift-invariant subspaces ofL ₂(R ^d ). Quasi-interpolants and cardinal interpolants are special operators of this type, and we give a complete characterization of the order in terms of some type of ellipticity condition for a related function. We apply these results by showing that theL ₂-approximation order of a closed shift-invariant subspace can often be realized by such an operator.     

Shift-invariant spaces on LCA groups

In this article we extend the theory of shift-invariant spaces to the context of LCA groups. We introduce the notion of H-invariant space for a countable discrete subgroup H of an LCA group G, and show that the concept of range function and the techniques of fiberization are valid in this context. As a consequence of this generalization we prove characterizations of frames and Riesz bases of these spaces extending previous results, that were known for Rd and the lattice Zd.     

Operators associated with soft and hard spectral edges from unitary ensembles

Using Hankel operators and shift-invariant subspaces on Hilbert space, this paper develops the theory of the integrable operators associated with soft and hard edges of eigenvalue distributions of random matrices. Such Tracy-Widom operators are realized as controllability operators for linear systems, and are reproducing kernels for weighted Hardy spaces, known as Sonine spaces. Periodic solutions of Hill's equation give a new family of Tracy-Widom type operators. This paper identifies a pair of unitary groups that satisfy the von Neumann-Weyl anti-commutation relations and leave invariant the subspaces of L² that are the ranges of projections given by the Tracy-Widom operators for the soft edge of the Gaussian unitary ensemble and hard edge of the Jacobi ensemble.     

On the hyperinvariant subspace problem III

In two recent papers (Foias and Pearcy, J. Funct. Anal., in press, Hamid et al., Indiana Univ. Math. J., to appear), the authors reduced the hyperinvariant subspace problem for operators on Hilbert space to the question whether every C₀₀-(BCP)-operator that is quasidiagonal and has spectrum the unit disc has a nontrivial hyperinvariant subspace (n.h.s.). In this note, we continue this study by showing, with the help of a new equivalence relation, that every operator whose spectrum is uncountable, as well as every nonalgebraic operator with finite spectrum, has a hyperlattice (i.e., lattice of hyperinvariant subspaces) that is isomorphic to the hyperlattice of a C₀₀, quasidiagonal, (BCP)-operator whose spectrum is the closed unit disc.     

On disjoint function systems and complemented subspaces

A sufficient condition for the complementability of subspaces generated by disjoint function systems in rearrangement invariant spaces is given. Orthogonal projections in L _p -spaces are extended to certain rearrangement invariant spaces. Applications to Lorentz spaces are given.     

Invariant subspaces for operator semigroups with commutators of rank at most one

Let X be a complex Banach space of dimension at least 2, and let S be a multiplicative semigroup of operators on X such that the rank of ST−TS is at most 1 for all {S,T}⊂S. We prove that S has a non-trivial invariant subspace provided it is not commutative. As a consequence we show that S is triangularizable if it consists of polynomially compact operators. This generalizes results from [H. Radjavi, P. Rosenthal, From local to global triangularization, J. Funct. Anal. 147 (1997) 443-456] and [G. Cigler, R. Drnovšek, D. Kokol-Bukovšek, T. Laffey, M. Omladi?, H. Radjavi, P. Rosenthal, Invariant subspaces for semigroups of algebraic operators, J. Funct. Anal. 160 (1998) 452-465].     

On the construction of multivariate (pre)wavelets

A new approach for the construction of wavelets and prewavelets onR ^d from multiresolution is presented. The method uses only properties of shift-invariant spaces and orthogonal projectors fromL ₂(R ^d ) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermore, this approach allows a simple derivation of previous, as well as new, constructions of wavelets, and leads to a complete resolution of questions concerning the nature of the intersection and the union of a scale of spaces to be used in a multiresolution.     

The structure of shift invariant spaces on a locally compact abelian group

We investigate shift invariant subspaces of L²(G), where G is a locally compact abelian group. We show, among other things, that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame.     

Characteristic functions and joint invariant subspaces

Let T:=[T₁,…,Tn] be an n-tuple of operators on a Hilbert space such that T is a completely non-coisometric row contraction. We establish the existence of a “one-to-one” correspondence between the joint invariant subspaces under T₁,…,Tn, and the regular factorizations of the characteristic function ΘT associated with T. In particular, we prove that there is a non-trivial joint invariant subspace under the operators T₁,…,Tn, if and only if there is a non-trivial regular factorization of ΘT. We also provide a functional model for the joint invariant subspaces in terms of the regular factorizations of the characteristic function, and prove the existence of joint invariant subspaces for certain classes of n-tuples of operators.We obtain criteria for joint similarity of n-tuples of operators to Cuntz row isometries. In particular, we prove that a completely non-coisometric row contraction T is jointly similar to a Cuntz row isometry if and only if the characteristic function of T is an invertible multi-analytic operator.     

Sufficient conditions for irregular Gabor frames

Finding general and verifiable conditions which imply that Gabor systems are (resp. cannot be) Gabor frames is among the core problems in Gabor analysis. In their paper on atomic decompositions for coorbit spaces [H.G. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations, and their atomic decomposition, I, J. Funct. Anal. 86 (1989), 307–340], the authors proved that every Gabor system generated with a relatively uniformly discrete and sufficiently dense time-frequency sequence will allow series expansions for a large class of Banach spaces if the window function is nice enough. In particular, such a Gabor system is a frame for the Hilbert space of square integrable functions. However, their proof is based on abstract analysis and does not give direct information on how to determine the density in the sense of directly applicable estimates. It is the goal of this paper to present a constructive version of the proof and to provide quantitative results. Specifically, we give a criterion for the general case and explicit density for some cases. We also study the existence of Gabor frames and show that there is some smooth window function such that the corresponding Gabor system is incomplete for arbitrary time-frequency lattices.     

Perturbation of frame sequences in shift-invariant spaces

We prove a new perturbation criteria for frame sequences, which generalizes previous results and is easier to apply. In the special case of frames infinitely generated shift-invariant subspaces of L²(ℝ^d) the condition can be formulated in terms of the norm of a finite Gram matrix and a corresponding rank condition.     

Ergodic theorems for tensor products

An ergodic theorem is proved for tensor products of Banach spaces. As a special case, an ergodic theorem is proved for vector-valued L^p-spaces. This theorem generalizes results of Aribaud, J. Funct. Anal.5 (1970), 395–411, and Dinculeanu, J. Funct. Anal.12 (1973), 229–235.     

Invariant subspaces for positive operators on locally convex solid Riesz spaces

We prove that an x₀-quasinilpotent semigroup S of continuous positive linear operators on a locally convex solid Riesz space X has a common invariant subspace. Using this, a result which implies the main theorem of Abramovich, Aliprantis and Burkinshaw [J. Funct. Anal. 115 (1993) 418424] is also given.     

Shift-invariant spaces of tempered distributions and Lp-functions

This paper studies the structure of shift-invariant spaces. A characterization for the univariate shift-invariant spaces of tempered distributions is given. In L_p case, an inclusive relation in terms of Fourier transform is established.     

Generalized intersection bodies are not equivalent

In [A. Koldobsky, A functional analytic approach to intersection bodies, Geom. Funct. Anal. 10 (2000) 1507-1526], A. Koldobsky asked whether two types of generalizations of the notion of an intersection body are in fact equivalent. The structures of these two types of generalized intersection bodies have been studied by the author in [E. Milman, Generalized intersection bodies, J. Funct. Anal. 240 (2) (2006) 530-567], providing substantial evidence for a positive answer to this question. The purpose of this note is to construct a counter-example, which provides a surprising negative answer to this question in a strong sense. This implies the existence of non-trivial non-negative functions in the range of the spherical Radon transform, and the existence of non-trivial spaces which embed in Lp for certain negative values of p.     

Frames, modular functions for shift-invariant subspaces and FMRA wavelet frames

We introduce the concept of the modular function for a shift-invariant subspace that can be represented by normalized tight frame generators for the shift-invariant subspace and prove that it is independent of the selections of the frame generators for the subspace. We shall apply it to study the connections between the dimension functions of wavelet frames for any expansive integer matrix and the multiplicity functions for general multiresolution analysis (GMRA). Given a frame mutiresolution analysis (FMRA), we show that the standard construction formula for orthonormal multiresolution analysis wavelets does not yield wavelet frames unless the underlying FMRA is an MRA. A modified explicit construction formula for FMRA wavelet frames is given in terms of the frame scaling functions and the low-pass filters.

     

Infinite matrices with “few” non-zero entries and without non-trivial invariant subspaces

In this paper we investigate a family of infinite matrices that act on ?₁. We derive a condition sufficient to guarantee that a matrix has no non-trivial closed invariant subspaces. As a result, a simplest known operator on ?₁ without invariant subspaces is obtained. All entries of the matrix of the example but one are non-negative.     

Subspace dual super wavelet and Gabor frames

Due to its potential applications in multiplexing techniques, the study of superframes has interested some researchers. This paper addresses dual super wavelet and Gabor frames in the subspace setting. We obtain a basic-equation characterization for subspace dual super wavelet and Gabor frames. In addition, applying this characterization, we derive a procedure that allows for constructing subspace dual super wavelet frames based on certain subspace dual super Gabor frames, and vice versa. Our results are new even in L²(R;C ^L ) setting.