Noncommutative Approximation: Inverse-Closed Subalgebras and Off-Diagonal Decay of Matrices

Abstract

We investigate two systematic constructions of inverse-closed subalgebras of a given Banach algebra or operator algebra A, both of which are inspired by classical principles of approximation theory. The first construction requires a closed derivation or a commutative automorphism group on A and yields a family of smooth inverse-closed subalgebras of A that resemble the usual Hölder-Zygmund spaces. The second construction starts with a graded sequence of subspaces of A and yields a class of inverse-closed subalgebras that resemble the classical approximation spaces. We prove a theorem of Jackson-Bernstein type to show that in certain cases both constructions are equivalent.These results about abstract Banach algebras are applied to algebras of infinite matrices with off-diagonal decay. In particular, we obtain new and unexpected conditions of off-diagonal decay that are preserved under matrix inversion.

What Is Variable Bandwidth?

“…The characterization (i) ⇔ (v) is proved in [34,45]. See also [15] for a related characterization of bandwidth. (ii) If Λ is compact, the Paley-Wiener space P W Λ (A) is a reproducing kernel Hilbert space; its kernel is…”

Decay Bounds for Functions of Hermitian Matrices with Banded or Kronecker Structure

“…A related but quite distinct line of research concerned the study of inverse-closed matrix algebras, where the decay behavior in the entries of a (usually infinite) matrix A is "inherited" by the entries of A −1 . Here we mention [33], where it was observed that a similar decay behavior occurs for the entries of f (A) = A −1/2 , as well as [2,3,26,27,35], among others.The study of the decay behavior for general analytic functions of banded matrices, including the important case of the matrix exponential, was initiated in [6,32] and continued for possibly nonnormal matrices and general sparsity patterns in [7]; further contributions in these directions include [4,16,38,42]. Collectively, these papers have largely elucidated the question of when one can expect exponential decay in the entries…”

“…The interest for the decay behavior of matrix functions stems largely from its importance for a number of applications, including numerical analysis [6,13,16,17,22,40,46], harmonic analysis [2,26,33], quantum chemistry [5,11,37,42], signal processing [35,43], quantum information theory [14,15,23], multivariate statistics [1], queuing models [9,10], control of large-scale dynamical systems [29], quantum dynamics [25], random matrix theory [41], and others. The first case to be analyzed in detail was that of f (A) = A −1 ; see [17,18,22,34].…”

Wiener's Lemma for Infinite Matrices of Gohberg-Baskakov-Sjöstrand Class

“…Schur class is not inverse-closed in B(ℓ 2 ) but the weighted Gröchenig-Schur class is when the weight satisfies the GRS-condition [2,6,7,9,17,19,22]. The Gohberg-Baskakov-Sjöstrand class [3,19,20].…”