A numerical range for two linear operators
Charles Amelin · Pacific Journal of Mathematics · 1973
A numerical range for two closed, linear operators is defined for the purpose of obtaining some new results on the stability of index of a Fredholm operator perturbed by a bounded or relatively bounded operator.()• Introduction* One of the objects of the present paper is to study the eigenvalue problem Tx = xAx by means of a two (linear) operator or "bioperative" numerical range.We recall that Toeplitz [30] defined the numerical range for matrices in 1918.Then Wintner [31] in 1930 and Stone [27], [28] in 1930 and 1932 discussed the relationship between the convex hull of the spectrum of a bounded linear operator on a Hubert space and its numerical range.In 1943, J. Dieudonne [4] and S. M. NikoPskii [24] laid the groundwork which would later help to show that the index of a bounded semi-Fredholm operator is stable under perturbation by a bounded linear operator of sufficiently small norm.This result was established in 1951 for bounded operators on a Hubert space by F.V.Atkinson [2] and (independently) I. C. Gohberg [9], [10], and [11].The following year, M. G. Krein and M. A. KrasnosePskii [20], B. Sz.-Nagy [29], and I. C. Gohberg [12] generalized these results to unbounded closed linear operators.M. G. Krein and M. A. KrasnosePskii also established the semi-stability of the nullity and deficiency.To understand the foundations and historical development of the whole theory, the reader is referred to the comprehensive article of Gohberg and Krein [13] which appeared in 1957.Among the many innovations appearing in the 1958 paper of T. Kato [18] was the concept of the "lower bound" (now called the "minimum modulus") of a linear operator A defined on a Banach space.The main reason for defining the minimum modulus of A was to obtain as small a disc about the origin as possible so that ind(Ύ-XA) = ind(A) for all λ outside that disc, A being semi-Fredholm and T being bounded or relatively bounded with respect to A. In this paper we introduce a bioperative numerical range which will improve that result for Fredholm operators defined on a Hubert space.The improvement is a consequence of the fact that the bioperative numerical range for Fredholm A and relatively bounded T defined on a Hubert space is always contained in the aforementioned disc and that the index of T -XA remains constant if λ is not in the closure of that bioperative numerical range.For another use of the bioperative numerical range, we recall