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Hyperexpansive composition operators. (English) Zbl 1066.47028

Let \(T: D(H)\to H\) be a linear map defined on a linear subspace \(D(H)\) of a Hilbert space \(H\) and let \[ \Theta_{T,n}(f)=\sum_{0\leq p\leq n}(-1)^p{n \choose p} | | T^pf| | ^2, \; f\in D(T^n),\quad n\geq 1. \] \(T\) is called \(k\)-hyperexpansive if \(\Theta_{T,n}(f)\leq 0\) for \(f\in D(T^n)\) and \(n=1,\dots,k\), and completely hyperexpansive if \(\Theta_{T,n}(f)\leq 0\) for \(f\in D(T^n)\) and all \(n\geq 1\). The author studies bounded and unbounded hyperexpansive composition operators on measure spaces \(L^2(\mu)\).

MSC:

47B33 Linear composition operators
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