Near-Riesz and pseudo-Riesz bases of Hilbert C*-modules and semi-C*-Fredholm operators

Abstract

Several useful concepts of almost frames or Riesz bases of Hilbert spaces gave us the motivation to review the theory of C*-module frames and C*-module Riesz bases for Hilbert C*-modules, searching for similar useful structures. We give a survey of the background in C*-module frame theory, summarizing and particularly extending or correcting earlier results. The concepts of near-Riesz bases, pseudo-frames, pseudo-Riesz sequences, and pseudo Riesz bases of Hilbert C*-modules are introduced and equivalently related to (semi-)C*-Fredholm properties of their synthesis operators. The notion of the excess of a C*-module frame is introduced, taking values in related C* algebras. Indicators for possible erasures of frame elements are described with the aim to preserve robustness of frames. Introducing a new notion of pseudo-Riesz-Weyl sequences in the standard Hilbert C*-module $l_2(A)$ we reverse the found correspondence by assuming the associated synthesis operator of a Bessel sequence to be (upper semi-)C*-Weyl. Furthermore, we deduce from the invariance of (upper semi-)C*-Fredholm and C*-Riesz-Weyl properties of synthesis operators with respect to ''compact'' perturbations the perturbation invariance of the respective classes of module Bessel sequences.

arXiv commentWe joined two further sections from the first version and worked on the presentation of the content, adding explanations, historical remarks and references. [The first two versions are still available on the arxiv.]