On the resolvent of a dilation for polynomial bounded operators (Q1341062)

Let \(H\) be a separable complex Hilbert space and let \(T\) be in \(L(H)\). A subspace \(M\subset H\) is said to be semi-invariant for \(T\) if there exist invariant subspaces \(N_ 1\supset N_ 2\) for \(T\) such that \(M= N_ 1\ominus N_ 2\). The author proves that if \(T\) is polynomially bounded, then, for every \(\varepsilon> 0\), there exists a Hilbert space \(K= K(\varepsilon)\) and a weakly centered polynomially bounded operator \(\widehat T= \widehat T(\varepsilon)\in L(K)\) whose spectrum is the unit circle such that \(H\) is semi-invariant for \(\widehat T\) and such that \(\|(\widehat T- \lambda)^{- 1}\|\leq M(1- | \lambda|)^{- \varepsilon- 3/2}\) for some \(M= M(\varepsilon)\) and every number \(\lambda\) with \(|\lambda|\neq 1\). It is also proven that every polynomially bounded operator is similar to a contraction iff every weakly centered polynomially bounded generalized scalar operator whose spectrum is the unit circle is similar to a contraction.