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PreprintCC BY-NC-ND 4.0arXiv

The reverse Ritz inclusion: computable bounds on the pseudospectra of an operator by those of a finite-rank compression

  • Isaia Nisoli
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Abstract

We consider a bounded operator on a separable complex Hilbert space, reached only through its action on vectors and that of its adjoint, and split it along a subspace of finite dimension into a compression, a far block acting on the complement, and the two couplings between them. Given upper bounds on the norms of the couplings and of the far block, we show that outside a disc a pseudospectrum of the operator lies in a pseudospectrum of the compression, with an explicit loss in the level; this reverses the classical inclusion of the Ritz values in a pseudospectrum of the operator. The same bounds, with the smallest singular value of the shifted compression, bound the resolvent of the operator, count its eigenvalues with algebraic multiplicity inside every contour outside the disc on which a computed inequality holds, and enclose its spectral projector; no normality, self-adjointness or positivity is used.

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Cite

BibTeXRIS
BibTeX
@misc{nisoli2026reverse,
  title = {{The reverse Ritz inclusion: computable bounds on the pseudospectra of an operator by those of a finite-rank compression}},
  author = {Isaia Nisoli},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30530},
  eprint = {2610.30530},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30530},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - The reverse Ritz inclusion: computable bounds on the pseudospectra of an operator by those of a finite-rank compression
AU  - Isaia Nisoli
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.30530
UR  - https://arxiv.org/abs/2610.30530
AB  - We consider a bounded operator on a separable complex Hilbert space, reached only through its action on vectors and that of its adjoint, and split it along a subspace of finite dimension into a compression, a far block acting on the complement, and the two couplings between them. Given upper bounds on the norms of the couplings and of the far block, we show that outside a disc a pseudospectrum of the operator lies in a pseudospectrum of the compression, with an explicit loss in the level; this reverses the classical inclusion of the Ritz values in a pseudospectrum of the operator. The same bounds, with the smallest singular value of the shifted compression, bound the resolvent of the operator, count its eigenvalues with algebraic multiplicity inside every contour outside the disc on which a computed inequality holds, and enclose its spectral projector; no normality, self-adjointness or positivity is used.
ER  -