The reverse Ritz inclusion: computable bounds on the pseudospectra of an operator by those of a finite-rank compression
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Nisoli, Isaia
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funders: Swedish Foundation for International Cooperation in Research and Higher Education, Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Conselho Nacional de Desenvolvimento Científico e Tecnológico, Universidade Federal do Rio de Janeiro, Engineering and Physical Sciences Research Council, Research Institute for Mathematical Sciences, Division of Mathematical Sciences
OpenAlex ID: W7223342977