A dilation theory for polynomially bounded operators
A bounded operator \(T \in {\mathcal L} ({\mathcal H})\) on a separable, infinite- dimensional complex Hilbert space \({\mathcal H}\) is called polynomially bounded \((T \in PB({\mathcal H}))\) if there exists a constant \(K>0\) such that \(\| p(T) \| \leq K \sup\{| p(z) |:| z |=1\}\) for every polynomial \(p\). It is an open problem whether every \(T \in PB({\mathcal H})\) is similar to a contraction. In this paper the author proves that (1) For every \(T \in PG({\mathcal H})\), tere exists a Hilbert space \({\mathcal K} \supset {\mathcal H}\) and an operator \(\hat T \in {\mathcal L} ({\mathcal K})\) such that (a) \(\hat T \in PB({\mathcal K})\), (b) \({\mathcal H}\) is a semi-invariant subspace for \(\hat T\) (i.e., there exist invariant subspaces \({\mathcal N}_ 1 \supset {\mathcal N}_ 2\) for \(\hat T\) such that \({\mathcal H}={\mathcal N}_ 1 \ominus {\mathcal N}_ 2\), (c) the spectrum of \(\hat T\) is the unit circle, and (d) \(\hat T\) is weakly centered (i.e., \(\hat T^*\hat T\) commutes with \(\hat T\hat T^*)\). (2) \(T \in PB({\mathcal H})\) is similar to a contraction if and only if every \(T_ 1 \in PB({\mathcal H})\) which satisfies conditions (c) and (d) above (with \(T_ 1\) in the place of \(\hat T)\) is similar to a contraction.
scientific article; zbMATH DE number 220669