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Logarithms and imaginary powers of closed linear operators

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Abstract

The imaginary powersA it of a closed linear operatorA, with inverse, in a Banach spaceX are considered as aC 0-group {exp(itlogA);t ∈R} of bounded linear operators onX, with generatori logA. Here logA is defined as the closure of log(1+A) − log(1+A −1). LetA be a linearm-sectorial operator of typeS(tan ω), 0≤ω≤(π/2), in a Hilbert spaceX. That is, |Im(Au, u)| ≤ (tan ω)Re(Au, u) foru ∈D(A). Then ω±ilog(1+A) ism-accretive inX andilog(1+A) is the generator of aC 0-group {(1+A)it;t ∈R} of bounded imaginary powers, satisfying the estimate ‖(1+A)it‖ ≤ exp(ω|t|),t ∈R. In particular, ifA is invertible, then ω±ilogA ism-accretive inX, where logA is exactly given by logA=log(1+A)−log(1+A −1), and {A it;t ∈R} forms aC 0-group onX, with the estimate ‖A it‖ ≤ exp(ω|t|),t ∈R. This yields a slight improvement of the Heinz-Kato inequality.

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  1. Department of Mathematics, Science University of Tokyo, 162-8601, Tokyo, Japan

    Noboru Okazawa

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Okazawa, N. Logarithms and imaginary powers of closed linear operators. Integr equ oper theory 38, 458–500 (2000). https://doi.org/10.1007/BF01228608

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