A characterization of bilateral operator weighted shifts being Cowen-Douglas operators (Q2731911)

A bilateral weighted shift of multiplicity \(n\) is an operator on \(\ell^2 (\ell^2 _n)\) of the form \(S: \{ x_j \}\mapsto \{ W_j x_{j-1} \}_{j\in{\mathbb Z}}\), where \(\{W_j \}\) is a uniformly bounded sequence of operators on \(l_n ^2\). A Hilbert space operator \(T:{\mathcal H}\to{\mathcal H}\) is called a Cowen-Douglas operator [see \textit{M. J. Cowen} and \textit{R. G. Douglas}, Acta Math. 141, 187-261 (1978; Zbl 0427.47016)] if there exists a connected open set \(\Omega \subset \sigma (T)\) and a natural number \(m\) such that \(\text{Im }(T-\lambda)={\mathcal H}\), and the kernels of \((T-\lambda)\) for \(\lambda \in \Omega\) are all of dimension \(m\) and their linear span is dense in \(\mathcal H\). It is shown that the above S is a Cowen-Douglas operator if and only if there exists \(\lambda _0\) such that \(S-\lambda _0\) is Fredholm and has index \(n\).