Operators which satisfy polynomial growth conditions (Q908493)

The author considers the class \({\mathcal P}\) of bounded linear operators S on a Banach space X for which the growth of \(\| \exp (itS)\|\) is at most polynomial in \(t\in {\mathbb{R}}\), explicitly: \(\| \exp (itS)\| \leq K(1+| t|^{\delta})\) for some \(K>0\) and \(\delta\geq 0\). It is shown that the operators in this class have many interesting properties in common with selfadjoint operators. In particular, for any S in \({\mathcal P}(X)\) the author proves the following properties: 1. The spectrum of S is real. 2. There exist \(K>0\) and \(\delta >0\) such that for all \(\lambda\in {\mathbb{C}}\) with Im(\(\lambda)\neq 0\), \(\| (\lambda -S)^{-1}\| \leq K(1+| Im(\lambda)|^{-\delta}).\) 3. For all \(\lambda\in {\mathbb{C}}\), \(\lambda\)-S has finite ascent. 4. The closed subalgebra generated by S and the identity is regular. 5. If the spectrum of S contains more than one number, then S has a proper closed hyper-invariant subspace. 6. If \(S,T\in {\mathcal P}(X)\) and \(ST=TS\), then \(S+T\in {\mathcal P}(X)\) and ST\(\in {\mathcal P}(X).\) The author also gives several types of operators that are in \({\mathcal P}(X)\).