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PreprintarXiv licencearXiv

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

  • Michael Frank
  • Stefan Ivkovic
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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 l2(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.

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Cite

BibTeXRIS
BibTeX
@misc{frank2026nearriesz,
  title = {{Near-Riesz and pseudo-Riesz bases of Hilbert C*-modules and semi-C*-Fredholm operators}},
  author = {Michael Frank and Stefan Ivkovic},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.16597},
  eprint = {2606.16597},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.16597},
}
RIS
TY  - UNPB
TI  - Near-Riesz and pseudo-Riesz bases of Hilbert C*-modules and semi-C*-Fredholm operators
AU  - Michael Frank
AU  - Stefan Ivkovic
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2606.16597
UR  - https://arxiv.org/abs/2606.16597
AB  - 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.
ER  -