Sinclair type inequalities for the local spectral radius and related topics (Q1105164)

Using a result of \textit{B. Ya. Levin} [Amer. Math. Soc. 1964, VIII, 493 p. (1964; Zbl 0152.067)] concerning the class P of entire functions, the author obtains a general pointwise inequality for the class of all bounded operators A on a Banach space X with real spectra such that \[ \| e^{itA}\| =O(| t|^ p)\text{ for some } p>0 \] when \(t\in {\mathbb{R}}\), \(| t| \to \infty.\) As a corollary of his result, he shows that \[ \| Ax\| \leq \| x\| r(A+iB,x), \] where A and B are commuting Hermitian operators on X and r(T,x) is the local spectral radius of T at x is defined as \[ r(T,x):=\limsup_{n\to \infty}\| T^ nx\|^{1/n}. \] In the particular case, when \(B=0\), he obtains the Sinclair lemma [Am. Math. Month. 78, 871-873 (1971)] for Hermitian operators on a Banach space.