\(N\)-hypercontractivity and similarity of Cowen-Douglas operators (Q1987004)

For a positive integer \(n\), an \(n\)-hypercontraction (see [\textit{J. Agler}, J. Oper. Theory 13, 203--217 (1985; Zbl 0593.47022)] for more details) is an operator \(T \in L(\mathcal{H})\). Here, we denote by \(L(\mathcal{H})\) the set of all bounded linear operators defined on the separable Hilbert space \(\mathcal{H}\) satisfying \[ \sum_{j=0}^{k} (-1)^{j}\binom{k}{j}(T^{*})^{j}T^{j} \ge 0 \] for all \( 0 \le k \le n\). Let \(\mathcal{H}_{w}^{2}\) be the weighted space which consists of all holomorphic functions \(f(z)=\sum_{j=0}^{\infty} a_{j}z^{j}\) satisfying \[ \sum_{j=0}^{\infty} |a_{j}|^{2}w_{j} < \infty. \] In this paper, the authors show the following. Theorem 1.1. Let \(T\) be the backward shift operator defined as \(T f(z)=zf(z)\) on the \(\mathcal{H}_{w}^{2}\), where \(w_{j}>0\), \(\liminf_{j \rightarrow \infty}|w_{j}|^{\frac{1}{j}}=1\) and \(\sup_{j}\frac{w_{j+1}}{w_{j}} < \infty\). If \(T\) is a \(n\)-hypercontraction, then for every nonnegative inter \(j\), we have \[ \frac{w_{j+1}}{w_{j}} \le \frac{1+j}{n+j}. \] Technically, the proof is based on two lemmas (Lemma 1.3 and 1.4) in which certain systems of linear equations is considered. The result is interesting, partially because researchers want to seek the ``geometric similarity invariant'' of Cowen-Douglas operators (see [\textit{R. G. Douglas} et al., J. Lond. Math. Soc., II. Ser. 88, No. 3, 637--648 (2013; Zbl 1290.47017); \textit{Y.-L. Hou} et al., Stud. Math. 236, No. 2, 193--200 (2017; Zbl 1450.47012)]). Moreover, the ``contraction'' assumption cannot be ignored besides the curvature condition when considering the similarity to the backward shift on Hardy space [\textit{H.-K. Kwon} and \textit{S. Treil}, Integral Equations Oper. Theory 66, No. 4, 529--538 (2010; Zbl 1204.47027)]. At the end of the paper, after proving a technical Lemma 2.5, Example 2.7 emphasizes the observation of [Kwon and Treil, loc.\,cit.]\ again.