Decomposition of operator functions and the multiplication problem for small ideals

Abstract

Let X be a Banach space, J$\subseteq$ L(X) a Fréchet ideal and G a region in ℂ^N. If J is non-locally-convex, then it is a problem whether A ∈H(G,L(X)), B ∈H(G,J) implies A·B,B·A ∈H(G,J). A positive answer to this question would sharpen an additive decomposition theorem of B. Gramsch [6] and B. Gramsch-W. Kaballo [8] for resolvents of semi-Fredholm functions. Here it is proved that if J is contained in another ideal J₁ such that the inclusion map i : J → J₁ is the product of N exponentially galbed maps in the sense of P. Turpin [18], [19], then A·B,B·A ∈H(G,J₁). An example shows that this is false if i is only a product of N-1 exponentially galbed maps. Thus a sharpening of the decomposition result mentioned above is obtained. Finally, for N=1, a sharper version of a multiplicative decomposition theorem of G.Ph.A. Thijsse [17] for FG-meromorphic functions is proved.