Strip-type Operators and the Logarithm

Abstract

As a straightforward abstraction of the logarithm of an injective sectorial operator we introduce the notion of a strip-type operator (Section 4.1). Since the resolvent of a strip-type operator by definition is bounded outside a horizontal strip, a functional calculus based on Cauchy integrals can be set up (Section 4.2). Section 4.3 is devoted to prove the main result, which states equality between the spectral angle of an injective sectorial operator A and the spectral height of the strip-type operator log A. As a corollary one obtains an important theorem of Prüss and Sohr, saying that in the case where A ∈ BIP, the group type of (A^is)_s∈ℝ is always larger than the spectral angle of A. In Section 4.4 the problem of ‘inversion’ is discussed, namely the question, which strip-type operators are actually logarithms of sectorial operators. Here we present a theorem of Monniaux, slightly generalised. In Section 4.5 we construct the example of an injective sectorial operator A ∈ BIP on a UMD space with the property that the group type of (A^is)_s∈ℝ is larger than π.