A characteristic operator function for the class of \(n\)-hypercontractions (Q2497931)

Let \(H\) be a complex, not necessarily separable Hilbert space. An operator \(T \in \mathcal{L}(H)\) is called \(n\)-hypercontraction if \(\sum_{k=0}^m (-1)^k\binom{m}{k}T^{*k}T^k\geq 0\) for all \(1\leq m \leq n.\) In particular, \(T\) is a contraction. For \(n>1\), these operators were introduced and characterized by \textit{J.~Agler} [J.~Oper.\ Theory 13, No.~2, 203--217 (1985; Zbl 0593.47022)]. He also showed that \(T\) is a subnormal contraction if and only if \(T\) is an \(n\)-hypercontraction for all \(n\). Let \(\mathcal E\) be another Hilbert space. For \(n\geq 1\), the weighted Bergman space A\(_n(\mathcal E)\) is defined as the Hilbert space of all \(\mathcal E\)-valued functions \(f(z)=\sum_{k\geq 0}a_kz^k\) with \(z\) in the unit disc \(\mathbb D\) and \(a_k \in \mathcal E,\) such that \[ | | f| | ^2=\sum_{k\geq 0}| | a_k| | ^2\frac{1}{\binom{k+n-1}{k}}<\infty. \] (A\(_1(\mathbb C)\) is the Hardy space, whereas A\(_2(\mathbb C)\) is the Bergman space.) The shift operator \(S_n\) on A\(_n(\mathcal E)\) is defined by \(S_n(f)(z)=zf(z)\) with \(z \in \mathbb D\). For an \(n\)-hypercontraction \(T\) of \(C_{0.}\) type, i.\,e., \(T^nx \to 0\) for all vectors \(x\), the author shows that there is an isometry \(V_n\) from \(H\) into A\(_n(\mathcal D_{n,T})\) (with \(\mathcal D_{1,T}\) the usual defect space) such that \(V_nT=S_n^*V_n\). Therefore \(V_n(H)\) is invariant under \(S_n^*\) and its orthogonal complement \(\mathcal I_{n,T}\) is invariant under \(S_n\). The corresponding wandering subspace is \(\mathcal E_{n,T}= \mathcal I_{n,T}\ominus S_n(\mathcal I_{n,T})\). The author proves many results. One of them is the following: A function \(f \in\) A\(_n(\mathcal D_{n,T})\) is in the wandering subspace \(\mathcal E_{n,T}\) if and only if \(f(z)=W_{n,T}(z)x\) for some vector \(x\in \mathcal D_{n,T}^*\), where \(W_{n,T}\) is an operator-valued analytic function in the unit disc whose values are operators between the defect spaces \(\mathcal D_{n,T}^*\) and \(\mathcal D_{n,T}\). The generalized characteristic function \(W_{n,T}\) is the usual Sz.--Nagy--Foias characteristic function when \(n=1\) [\textit{B.~Sz.--Nagy} and \textit{C.~Foias}, ``Harmonic analysis of operators on Hilbert spaces'' (Budapest: Akadémiai Kiadó; Amsterdam-London: North--Holland) (1970; Zbl 0201.45003)].