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.
Highlights
The spectral properties of operator functions play an important role in mathematical analysis and in many applications [4,13,20]
The norm of the resolvent in a point ω is under some conditions bounded by a quantity that depend on the distance from ω to the numerical range [16]
Knowledge of the numerical range is important in perturbation theory and in several other branches of operator theory [12]
Summary
The spectral properties of operator functions play an important role in mathematical analysis and in many applications [4,13,20]. In this paper we introduce an enclosure of the numerical range of a class of rational operator functions whose values are linear operators in a Hilbert space H. This new enclosure is applicable in the infinite dimensional case as well as in the finite dimensional case. The derived enclosure of the pseudonumerical range provides a computable upper bound of the norm of the resolvent in the complement of the new enclosure of the numerical range. Our main results are Theorem 4.3, which shows how the boundary of the enclosure of the pseudospectra can be determined and Corollary 4.6 gives an estimate of the resolvent of (1.1).
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