Invariant Cocycles Have Abelian Ranges

 Let R be a discrete nonsingular equivalence relation on a standard probability space , and let V be an ergodic strongly asymptotically central automorphism of R. We prove that every V-invariant cocycle with values in a Polish group G takes values in an abelian subgroup of G. The hypotheses of this result are satisfied, for example, if A is a finite set, a closed, shift-invariant subset, V is the shift, μ a shift-invariant and ergodic probability measure on X, the two-sided tail-equivalence relation on X, a shift-invariant subrelation which is μ-nonsingular, and a shift-invariant cocycle.     

Invariant Cocycles Have Abelian Ranges

 Let R be a discrete nonsingular equivalence relation on a standard probability space , and let V be an ergodic strongly asymptotically central automorphism of R. We prove that every V-invariant cocycle with values in a Polish group G takes values in an abelian subgroup of G. The hypotheses of this result are satisfied, for example, if A is a finite set, a closed, shift-invariant subset, V is the shift, μ a shift-invariant and ergodic probability measure on X, the two-sided tail-equivalence relation on X, a shift-invariant subrelation which is μ-nonsingular, and a shift-invariant cocycle. (Received 15 September 2001)     

Weak Operator Topology, Operator Ranges and Operator Equations via Kolmogorov Widths

Let K be an absolutely convex infinite-dimensional compact in a Banach space χ. The set of all bounded linear operators T on χ satisfying TK ⊃ K is denoted by G(K). Our starting point is the study of the closure WG(K) of G(K) in the weak operator topology. We prove that WG(K) contains the algebra of all operators leaving [`(lin(K))]\overline{{\rm lin}(K)} invariant. More precise results are obtained in terms of the Kolmogorov n-widths of the compact K. The obtained results are used in the study of operator ranges and operator equations.     

About Operator Weighted Shift

Abstract Let H be a complex seperable Hilbert space and ~（Jt~） denote the collection of bounded linear operators on H. For an operator in L（H）, rad{A}＇ denotes the Jacobson radical of the commutant of A. This paper characterizes the similarity of strongly irreducible operator weighted shift in terms of {A}＇/rad{A}＇. Moreover, we suggest some ways to determine when an operator weighted shift is strongly irreducible and when its commutant is commutative.     

Operator Ranges in Banach Spaces,I

Let X be a Banach space. A subspace L of X is called an operator range if there exists a continuous linear operator T defined on some Banach space Y and such that TY = L. If Y = X then L is called an endomorphism range. The paper investigates operator ranges under the following perspectives: (1) Existence (Section 3), (2) Inclusion (Section 4), and (3) Decomposition (Section 5). Section 3 considers the existence in X of operator ranges satisfying certain conditions. The main result is the following: if X and Fare separable Banach spaces and T : Y → X is a continuous operator with nonclosed range, then there exists a nuclear operator R:Y→X such that T + R is injective and has nonclosed dense range (Theorem 3.2). Section 4 seeks to determine conditions under which every nonclosed operator range in a Banach space is contained in the range of some injective endomorphism with nonclosed dense range. Theorem 4.3 contains a sufficient condition for this. Examples of spaces satisfying this condition are c₀, l_p (1 < p < ∞), L_q (1 < q < 2) and their quotients. In particular, this answers a question posed by W. E. Longstaff and P. Rosenthal (Integral Equations and Operator Theory 9 , (1986), 820-830. Section 5 discusses the possibility of representing a given dense nonclosed operator range as the sum of a pair L_1, L₂ of operator ranges with zero intersection in the cases where (a) L₁ and L₂ are dense, (b) L₁ and L₂ are closed. The results generalize corresponding results, for endomorphisms in Hilbert space, of J. Dixmier (Bull. Soc. Math. France 77 (1949), 11-101 and P. A. Fillmore and J. P. Williams (Adv. Math. 7 (1971), 254-281. A final section is devoted to open problems.     

Small Transitive Families of Dense Operator Ranges

In complex, separable, infinite-dimensional Hilbert space there exist 5 proper dense operator ranges with the property that every operator leaving each of them invariant is a scalar multiple of the identity. The algebra of operators leaving a pair of proper dense operator ranges invariant can have an infinite nest of invariant subspaces. A slight extension of Foiaş' Theorem shows that it can also have a non-trivial reducing subspace. Submitted: July 13, 2001? Revised: December 6, 2001.     

Embeddings, Operator Ranges, and Dirac Operators

We present a generalized operator range construction associated to an indefinite unbounded selfadjoint operator that yields closed embeddings of Kreĭn spaces. As an application we obtain an energy space representation, in the sense of Friedrichs, of a general free Dirac operator.     

Wavelet Decompositions of Nonrefinable Shift Invariant Spaces

The motivation for this work is a recently constructed family of generators of shift invariant spaces with certain optimal approximation properties, but which are not refinable in the classical sense. We try to see whether, once the classical refinability requirement is removed, it is still possible to construct meaningful wavelet decompositions of dilates of the shift invariant space that are well suited for applications.     

p-Frames and Shift Invariant Subspaces of Lp

In this article we investigate the frame properties and closedness for the shift invariant space V_p(F) = { ?_i=1^r ?_{j ? \Zd} d_i(j) f_i (·-j):  ( d_i(j) )_{j ? \Zd} ? l^p }, \q 1 ￡ p ￡ ￥ . \displaystyle V_p(\Phi) = \left\{ \sum_{i=1}^r \sum_{j\in \Zd} d_i(j) \phi_i (\cdot-j): \ \left( d_i(j) \right)_{j\in \Zd}\in \ell^p \right\}, \q 1\le p \le \infty~. We derive necessary and sufficient conditions for an indexed family {f_i(·-j): 1 ￡ i ￡ r, j ? \Zd}\{\phi_i(\cdot-j):\ 1\le i\le r, j\in \Zd\} to constitute a pp-frame for V_p(F)V_p(\Phi), and to generate a closed shift invariant subspace of L^pL^p. A function in the L^pL^p-closure of V_p(F)V_p(\Phi) is not necessarily generated by l^p\ell^p coefficients. Hence we often hope that V_p(F)V_p(\Phi) itself is closed, i.e., a Banach space. For p 1 2p\ne 2, this issue is complicated, but we show that under the appropriate conditions on the frame vectors, there is an equivalence between the concept of pp-frames, Banach frames, and the closedness of the space they generate. The relation between a function f ? V_p(F)f \in V_p(\Phi) and the coefficients of its representations is neither obvious, nor unique, in general. For the case of pp-frames, we are in the context of normed linear spaces, but we are still able to give a characterization of pp-frames in terms of the equivalence between the norm of ff and an l^p\ell^p-norm related to its representations. A Banach frame does not have a dual Banach frame in general, however, for the shift invariant spaces V_p(F)V_p(\Phi), dual Banach frames exist and can be constructed.     

Spectral Analysis of a Weighted Shift Operator with Unbounded Operator Coefficients

We obtain theorems on the structure of the resolvent of a weighted shift operator with unbounded operator coefficients which acts in Banach spaces of two-sided sequences of vectors.     

Convex Potentials and Optimal Shift Generated Oblique Duals in Shift Invariant Spaces

We introduce extensions of the convex potentials for finite frames (e.g. the frame potential defined by Benedetto and Fickus) in the framework of Bessel sequences of integer translates of finite sequences in \(L^2(\mathbb {R}^k)\). We show that under a natural normalization hypothesis, these convex potentials detect tight frames as their minimizers. We obtain a detailed spectral analysis of the frame operators of shift generated oblique duals of a fixed frame of translates. We use this result to obtain the spectral and geometrical structure of optimal shift generated oblique duals with norm restrictions, that simultaneously minimize every convex potential; we approach this problem by showing that the water-filling construction in probability spaces is optimal with respect to submajorization (within an appropriate set of functions) and by considering a non-commutative version of this construction for measurable fields of positive operators.     

Dilations of Some VH-Spaces Operator Valued Invariant Kernels

We investigate VH-spaces (Vector Hilbert spaces, or Loynes spaces) operator valued Hermitian kernels that are invariant under actions of *-semigroups from the point of view of generation of *-representations, linearizations (Kolmogorov decompositions), and reproducing kernel spaces. We obtain a general dilation theorem in both Kolmogorov and reproducing kernel space representations, that unifies many dilation results, in particular B. Sz.-Nagy??s and Stinesprings?? dilation type theorems.     

Backward Aluthge Iterates of a Hyponormal Operator Have Scalar Extensions

The backward Aluthge iterate (defined below) of a hyponormal operator was initiated in [11]. In this paper we characterize the backward Aluthge iterate of a weighted shift. Also we show that the backward Aluthge iterate of a hyponormal operator has an analogue of the single valued extension property for . Finally, we show that backward Aluthge iterates of a hyponormal operator have scalar extensions. As a corollary, we get that the backward Aluthge iterate of a hyponormal operator has a nontrivial invariant subspace if its spectrum has interior in the plane.     

Complex Wavelets for Shift Invariant Analysis and Filtering of Signals

This paper describes a form of discrete wavelet transform, which generates complex coefficients by using a dual tree of wavelet filters to obtain their real and imaginary parts. This introduces limited redundancy (2^m:1 for m-dimensional signals) and allows the transform to provide approximate shift invariance and directionally selective filters (properties lacking in the traditional wavelet transform) while preserving the usual properties of perfect reconstruction and computational efficiency with good well-balanced frequency responses. Here we analyze why the new transform can be designed to be shift invariant and describe how to estimate the accuracy of this approximation and design suitable filters to achieve this. We discuss two different variants of the new transform, based on odd/even and quarter-sample shift (Q-shift) filters, respectively. We then describe briefly how the dual tree may be extended for images and other multi-dimensional signals, and finally summarize a range of applications of the transform that take advantage of its unique properties.     

On an Elliptic Differential-Difference Equation with Nonsymmetric Shift Operator

Mathematical Notes - An essentially nonlinear equation containing the product of the p-Laplacian and a nonsymmetric difference operator is considered. Sufficient conditions guaranteeing the...     

The Spectral Shift Function for Certain Block Operator Matrices

Let L₀ be a 2 × 2 diagonal self‐adjoint block operator matrix with entries A and D. If operators B and B* are added to the off diagonal zeros, certain parts of the spectrum of L₀ move to the right and other parts move to the left. In this paper it is shown that, correspondingly, if B is a trace class operator M. G. Krein's spectral shift function is of constant sign on certain intervals.