The finite fibre problem and an index formula for elementary operators

Abstract:

Let $J \subset L(K,H)$ be an operator ideal between Hilbert spaces, and let $S \in L{(H)^n},T \in L{(K)^n}$ be commuting tuples of continuous linear operators. It is shown that the elementary operator $R:J \to J,A \to \sum \nolimits _{i = 1}^n {{S_i}A{T_i}}$ determined by S and T satisfies the finite fibre property. As a consequence it follows that an index formula proved by L. Fialkow for elementary operators under the additional assumption of the finite fibre property holds true in general.