Local spectral properties of generators of \(C_0\)-groups and Bernstein type inequalities
In this paper, the authors study the local spectral properties of generators of \(C_0\) groups. Let \(T=\{T(t)\}_{t\in\mathbb R}\) be a \(C_0\) group on a complex Banach space \(X\) dominated by a weight function \(\omega(t)=(1+|t|)^{\alpha}\) (\(0\leq \alpha \leq 1\)) and let \(A\) be its generator with domain \(D(A)\). Among other things, it is proved that, if the operator \(A\) has compact local spectrum at a point \(x\in X\), then \(x\in D(A)\) and there exist double sequences of real numbers \((c_n)_{n\in\mathbb Z}\) and \((t_n)_{n\in\mathbb Z}\) such that \[ Ax=\sum_{n\in\mathbb Z}c_nT(t_n)x, \] where \( \sum_{n\in\mathbb Z}|c_n|=r_A(x)\), the local spectral radius of \(A\) at \(x\). As applications, the authors give several inequalities of Bernstein type in \(L^p\)-spaces.
scientific article; zbMATH DE number 5568991