Generalized Cowen-Douglas operators over Hilbert \(C^*\)-modules (Q1892613)

A Hilbert \(C^*\)-module is a mathematical object like a Hilbert space provided that the inner product takes its values in a general \(\mathbb{C}^*\)-algebra instead of being complex-valued. In this paper, the author generalizes to Hilbert \(C^*\)-modules the study of a class of bounded linear operators introduced on separable Hilbert space by M. Cowen and R. Douglas, operators defined as follows: Let \(\Omega\) be a non-empty bounded connected open subset of \(\mathbb{C}\). A bounded linear operator \(T\) on a Hilbert space \(H\) belongs to \(B_n (\Omega)\), \(n\) a positive integer, if, for \(\omega\in \Omega\), \(T-\omega\) is not invertible, \(\dim \ker (T-\omega) =n\), \(\text{ran} (T- \omega)= H\) and \(\{\ker (T- \omega)\), \(\omega\in \Omega\}\) generate \(H\). In particular, the author shows that, for \(n=1\), as in the classical Hilbert space case, such an operator in a Hilbert \(C^*\)-module is unitarily equivalent to the adjoint of the operator of multiplication by a coordinate function over some functional Hilbert module. The author shows also that every \(B_n (\Omega)\)-operator on a Hilbert space can be thought as a \(B_1 (\Omega)\)-operator on a related Hilbert \(C^*\)-module.