Some estimates for analytic functions of strip or sectorial operators (Q1423832)

Let \(H\) be a complex Hilbert space and \(T\) be a closed linear operator on \({\mathcal H}\) such that its domain \(D(T)\) is densely imbedded in \(H\). Consider the numerical range \(W(T)= \{\langle Tu,u\rangle: u\in D(T),\| u\|= 1\}\). If \(S\) is a strip or a sector of the complex plane such that \(W(T)\subset S\), then the authors provide an estimate of the norm of \(f(T)\) with respect to this uniform bound, where \(f\) is an analytic function that is uniformly bounded on \(S\).