The Fredholm index of quotient Hilbert modules (Q812537)

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).