The Fredholm index of quotient Hilbert modules
Let \(H\) be a Hilbert module in the sense of a Hilbert space equipped with a module structure over the polynomial ring \(A = C[z_1, \dots , z_n]\) in \(n\) complex variables such that the action by each coordinate function \(z_i\) induces a bounded operator on \(H\). Let \(I = (z_1, \dots , z_n)\) be the maximal ideal of \(A\) at the origin. If \(\dim H/IH < \infty\), then it is known that \(\dim H/I^kH < \infty\) for all \(k \in \mathbb N\), that the limit \[ e(H) = n! \lim_{k \to \infty} \frac{\dim H/I^kH}{k^n} \] exists and is an integer, called the Samuel multiplicity of \(H\); see \textit{X.~Fang} [Adv.\ Math.\ 186, No.~2, 411--437 (2004; Zbl 1070.47007)]. The author of the paper under review proves, for some certain classes of Hilbert modules, that the Fredholm index of a submodule is equal to its fibre dimension (an analytic notion) if and only if the Fredholm index of the quotient module is equal to its Samuel multiplicity (an algebraic notion).
scientific article; zbMATH DE number 5001039