Approximation of approximation numbers by truncation

On the Solvability Complexity Index, the 𝑛-pseudospectrum and approximations of spectra of operators

“…The literature is vast on spectral approximation and we can only refer to a subset here. For selected papers and books we consider to be important and related to the topic of this paper we refer to [9,11,10,26,32,15,14,41,19,18,28,42,47] for a functional analysis exposition and [50,48,25,39] for a more applied mathematical treatment.…”

Spectral Approximation of Bounded Self-Adjoint Operators—A Short Survey

“…Now we prove the following result from [4] which is the major tool that is used frequently in this note.…”

“…However, the inclusion in Theorem 2.9 helps us to determine the spectrum, with an additional assumption of connectedness of the essential spectrum. The details of this claim are given below, which is a brief review of the article [4] with some slight modifications. This will play a key role in the forthcoming sections.…”

“…So far there is no evidence of remainder estimation and the error estimation in these approximations in the case of an arbitrary self-adjoint operator to the best of our knowledge. The subsequent theorem taken from [4] denies the existence of spurious eigenvalues (points in those are not part of the spectrum) under the assumption of connectedness of essential spectrum.…”

“…To quote some of the recent contributions in this direction are due to W.B. Arveson [1], Albrecht Böttcher et al [4], E.B. Davies et al [6,7], I. Gohberg et al [9], A. Hansen [11], etc.…”