Equivalence, linearization, and decomposition of holomorphic operator functions

Abstract

This paper discusses the problem of classifying holomorphic operator functions up to equivalence. A survey is given in §1 of the main results about equivalence classes of holomorphic matrix functions and holomorphic Fredholm-operator functions. In §2, it is shown that given a holomorphic function A on a bounded domain Ω into a space of bounded linear operators between two Banach spaces, it is possible to extend the operators A( λ) (for each λ ϵ Ω) by an identity operator I Z in such a way that the extended operator function A(·) ⊕ I Z is equivalent on Ω to a linear function of λ, T − λI. Other versions of this “linearization by extension” are described, including the cases of entire functions and polynomials (where Ω = C ). As an application of these results, we consider the operator function equation A 2(λ) Z 2(λ) + Z 1(λ) A 1(λ) = C(λ), λ ϵ Ω , (∗) and explicitly construct the solutions Z 1 and Z 2. The formulas for Z 1 and Z 2 seem to be new, even when A 1, A 2 and C are matrix polynomials. The existence of solutions of ( ∗) makes it possible to analyze an operator function A whose spectrum decomposes into pairwise disjoint compact subsets σ 1, …, σ n of Ω. In this case, a suitable extension of A is equivalent on Ω to a direct sum of operator functions, A 1, …, A n , such that the spectrum of A i is σ i ( i = 1, …, n). In the final section of the paper, we discuss the relation between local and global equivalence on Ω, and show that there exist operator functions A and B which are locally equivalent on Ω, but admit no extensions (of the sort considered in this paper) which are globally equivalent on Ω.