Stability properties of finite meromorphic operator functions. II

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

• Publisher: Elsevier Science
• Year: 1974
• Weight: 558 KB
• Volume: 77
• Category: Article
• ISSN: 1385-7258
• DOI: 10.1016/1385-7258(74)90040-7

✦ Synopsis

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 space Z is complemented (in 2) if M is closed and there exists a closed subspace N of 2 such that Z=M @ N. 3.1 LEMMA. Let M and N be subspaces of X such that N C M and dim M/N -CO and the coefficients in the principal part of the Laurent expansion of A at & are all degenerate operators (that is, operators with Cnite-dimensional range spaces). Observe that any diagonal operator function at J,,J is finite meromorphic at Ao. Finite meromorphic operator functions which have the extra property that the constant term in the Laurent expansion at ilo is a Fredholm operator have been studied by several authors (see [l], [2], [a], [6], [8], PI, WI, WI, [24l and 1251). A h c aracterization of such operator functions in terms of the spaces HO and & is given at the end of this section. Other