A note on coherent states related to weighted shifts
If \(W\) is a weighted shift with positive weights and the coherence radius of \(W\), \(r(W)> 0\), then the authors show that \(h(z)\in D(W^*)\), the domain of the adjoint \(W^*\) of \(W\), where \(h(z)= e_ 0+ \sum^ \infty_{n= 1} {z^ n\over w_ 0\cdot\cdots\cdot w_{n-1}} e_ n\) with \(e_ n(z)= {z^ n\over \sqrt{n!}}\) for \(n\geq 0\), \[ z\in S(W)= \{z\in \mathbb{C}: | z|< r(W)\}. \] Also \(W^* h(z)= zh(z)\), \(z\in S(W)\) and \(\dim\text{Ker}(zI- W^*)= 1\), \(z\in S(W)\). Next, if \(W\) is a subnormal weighted shift with positive weights and if \(\mu(\mathbb{C}- S(W))= 0\), then the map \(f\to \widehat f\) is a unitary isomorphism of a separable Hilbert space \(H\) onto the set \(B^ 2(\mu)\) of all entire functions belonging to \(L^ 2(\mu)\). Furthermore, the identity operator \(I\) on \(H\) is given by \(I= \int_{S(W)} P_ z\| h(\bar z)\|^ 2 d\mu(z)\). Here, given \(z\in S(W)\), \(P_ z\) denotes the orthogonal projection of \(H\) onto the one-dimensional space spanned by the vector \(h(z)\).
subnormal weighted shift with positive weights
scientific article; zbMATH DE number 597457