On holomorphic relative inverses of operator-valued functions

Simeon Ivanov

Vol. 78 (1978), No. 2, 345–358

DOI: 10.2140/pjm.1978.78.345

Abstract

Let G be a complex domain, X and Y Banach spaces and A : G → L(X,Y ) holomorphic with Ker A(λ), Im A(λ) complemented, λ ∈ G. It is shown that the following conditions are equivalent: (1) A has a holomorphic relative inverse on G; (2) the function λ → Ker A(λ) is locally holomorphic on G; (3) the function λ → Im A(λ) is locally holomorphic on G. Based on this, it is shown that a semi-Fredholm-valued holomorphic function A has a holomorphic relative inverse on G if and only if dimKer A(λ) [codim Im A(λ), respectively] is constant on G.

The latter result is a generalization of the well-known result of Allan on one-side holomorphic inverses.

Mathematical Subject Classification 2000

Primary: 47A05

Secondary: 30G30

Milestones

Received: 4 November 1977

Published: 1 October 1978