FROM FINITE TO INFINITE DIMENSIONS: CHANCES AND CHALLENGES IN SPECTRAL THEORY

CHRISTIANE TRETTER

This lecture focuses on chances and challenges in obtaining reliable information on eigenvalues and, more generally, spectra of linear operators. Two aspects will be addressed. First, finite dimensional tools to enclose spectra of infinite dimensional problems will be presented. Spectral bounds in terms of these so-called block numerical ranges [1] improve classical numerical range bounds, both in infinite and finite dimensions. Secondly, infinite dimensional tools to capture spurious eigenvalues of finite dimensional spectral approximations will be showcased. These so-called essential numerical ranges [2], [3], originally designed to enclose essential spectra, turn out to be powerful tools to assess the reliability of finite dimensional spectral approximations for unbounded linear operators. Examples and applications illustrate the abstract results.

References

[1]    C. Tretter. Spectral theory of block operator matrices and applications, Imperial College Press, London, 2008.

[2]    S. Bögli, M. Marletta, C. Tretter. The essential numerical range for unbounded linear operators. J. Funct. Anal., 279 (2020), p. 49. Id/No 108509.

[3]    N. Hefti, C. Tretter. The essential numerical range for unbounded linear operators. Studia Math., 264 (2022), no. 3, 305–333.

University of Bern, Switzerland