Similarity to the backward shift operator on the Dirichlet space (Q2835252)

In this paper, the author gives a necessary condition for a Cowen-Douglas operator to be similar to the backward shift operator \(D_{n}^{\ast}\) on the Dirichlet space. The main theorem is the following (Theorem 2.2). Let \(H\) be a Hilbert space. Let \(D\) the unit disk. Suppose \(T:H\rightarrow H\) is a bounded linear operator that satisfies the following four properties: (1) \(\sum_{k=0}^\infty \frac{\| T^k\|^2}{k+1}\leq 1\). (2) The closed linear span of \(\bigcup_{\lambda \in D}\ker (T-\lambda I)=H\). (3) The subspaces \(\ker(T-\lambda I)\) analytically depend on \(\lambda\in D\). (4) There is an \(n\in N\) such that \(\dim(\ker(T-\lambda I))=n\) for all \(\lambda\in D\). If \(T\) is similar to the backward shift operator \(D_{n}^{\ast}\), then for some bounded subharmonic function \(\phi :D\rightarrow C\) we have NEWLINE\[NEWLINE\Delta \phi (\lambda)\geq \left \| \frac{\partial \Pi (\lambda)}{\partial \lambda}\right \|_{G_{2}}^{2}+\ell (\lambda)\quad \text{ for all \(\lambda \in D\),}NEWLINE\]NEWLINE where \(\Delta =\frac{1}{4}t(\frac{\partial ^{2}}{\partial x^{2}}+\frac{\partial^ {2}}{\partial y^{2}})\) is the normalized Laplacian, \(\Pi (\lambda):H \rightarrow \ker (T-\lambda I)\) is the orthogonal projection onto \(\ker (T-\lambda I)\), \(\| \cdot \|_{G_{2}}^{2}\) denotes the Hilbert-Schmidt operator norm, and NEWLINE\[NEWLINE l(\lambda)=\begin{cases} \frac{n[\log (1-|\lambda |^{2})+|\lambda |^{2}]} {[\log (1-|\lambda |^{2})(1-|\lambda |^{2})]^{2}}&\text{if } | \lambda |>0, \\ -\frac{n}{2} & \text{if }\lambda =0.\end{cases} NEWLINE\]