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Stability properties of finite meromorphic operator functions. I

✍ Scribed by H Bart; M.A Kaashoek; D.C Lay

Publisher
Elsevier Science
Year
1974
Weight
756 KB
Volume
77
Category
Article
ISSN
1385-7258

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✦ Synopsis

Let A be a meromorphic function with values in the space of bounded linear operators between two Banach spaces X and Y, and assume that the coefficients of the prinoipal part of the Laurent expansion of A at a certain point & are degenerate operators. In this paper it is shown that under rather general hypotheses the null spaces (resp. ranges) of A(1) converge in the gap topology to a certain subspace of X (reap. Y) as il approaches &. Further, under slightly stronger conditions, the null spaces (reap. ranges) of A(A) have E fixed complementary subspace in X (resp. Y) for all rl in some deleted neighbourhood of k. The hypotheses of these stability theorems are fulfilled if A is Fredholm at & or haa values in the set of degenerate operators.

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## Reduction to the holomorphic case We begin with two lemmas which will be useful both here and in other sections. The proof of Lemma 3.1(b) follows immediately from Theorem IV.1.12 in [l l] ; the proof of the other statements are elementary and will be omitted. We say that a subspace M of a Banach

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## The conjugate operator function In this section we study some important relations between the H, and K,,, spaces for A and those for the conjugate operator function A\*. We denote the conjugate spaces of X and Y by X\* and Y\*, respectively, and we define A\* to be the function with values in 9