A characterization of the generators of analytic C0‐semigroups in the class of scalar type spectral operators

Marat V Markin

doi:10.1155/s1085337504403054

Abstract

In the class of scalar type spectral operators in a complex Banach space, a characterization of the generators of analytic C0‐semigroups in terms of the analytic vectors of the operators is found.

Highlights

Let A be a linear operator in a Banach space X with norm ·
Each of equalities (1.4), the operator exponentials etA, t > 0, defined in the sense of the o.c. for scalar type spectral operators, is necessary and sufficient for A to be the generator of an analytic C0-semigroup
We consider the general of A being a generator of an analytic C0-semigroup {etA | t ≥ 0} in a complex Banach space X, without the assumption of A being a scalar type spectral operator

Summary

Introduction

As was shown in [13], a scalar type spectral operator A in a complex Banach space X generates an analytic C0-semigroup, if and only if, for some real ω and 0 < θ ≤ π/2, σ(A) ⊆ λ ∈ C | arg(λ − ω) ≥ π + θ , 2. Each of equalities (1.4), the operator exponentials etA, t > 0, defined in the sense of the o.c. for scalar type spectral operators, is necessary and sufficient for A to be the generator of an analytic C0-semigroup. We consider the general of A being a generator of an analytic C0-semigroup {etA | t ≥ 0} in a complex Banach space X, without the assumption of A being a scalar type spectral operator. As is seen, the negation of the fact that A generates an analytic C0semigroup implies that for any b > 0, the set σ(A) \ λ ∈ C | Re λ ≤ −b| Im λ|

Re λ

Let f