Enclosure of the Numerical Range of a Class of Non-selfadjoint Rational Operator Functions

Abstract

Abstract. In this paper we introduce an enclosure of the numerical range of a class of rational operator functions. In contrast to the numerical range the presented enclosure can be computed exactly in the infinite dimensional case as well as in the finite dimensional case. Moreover, the new enclosure is minimal given only the numerical ranges of the operator coefficients and many characteristics of the numerical range can be obtained by investigating the enclosure. We introduce a pseudonumerical range and study an enclosure of this set. This enclosure provides a computable upper bound of the norm of the resolvent.Mathematics Subject Classification. 47J10, 47A56, 47A12.

Enclosure of the numerical range and resolvent estimates of non-selfadjoint operator functions

“…The presented enclosure of the numerical range is optimal given the numerical ranges of A and B since for each possible pair W pAq and W pBq, there exist operators A and B such that the enclosure coincides with the numerical range of T . Although [ET17] considers a very special case, we will show in this paper that the main results for (1.1) also hold in a much more general setting. We will study operator functions with an arbitrary number of possibly unbounded operator coefficients.…”

Spectral properties of conservative, dispersive, and absorptive photonic crystals

“…In Engström and Torshage, we introduced an en enclosure of the numerical range W ( k ), which then also is an enclosure of the point spectrum. The construction of this enclosure is outlined below.…”

“…This enclosure of the numerical range is optimal given only the numerical ranges , W ( B 1,1 B 1,1∗) and it is of vital importance for the proofs of our results. In Engström and Torshage the set W Ω ( k ) is defined for the Riemann sphere C¯ instead of C but the proofs and the results are the same.…”

“…Since no other properties of A (0) k apart from self-adjointness were used, (47) holds for any self-adjoint operator A (0) k . In Engström and Torshage, [33] we introduced an en enclosure of the numerical range W( k ), which then also is an enclosure of the point spectrum. The construction of this enclosure is outlined below.…”

On Equivalence and Linearization of Operator Matrix Functions with Unbounded Entries

“…Important examples are systems of partial differential equations with λ-dependent coefficients or boundary conditions [1,9,10,19,23]. A concept of equivalence can be used to compare spectral properties of different operator functions and the problem of classifying bounded analytic operator functions modulo equivalence has been studied intensely [6,7,11,15].…”