Integral Equations and Operator Theory

, Volume 38, Issue 4, pp 458–500 | Cite as

Logarithms and imaginary powers of closed linear operators

  • Noboru Okazawa


The imaginary powersAit of a closed linear operatorA, with inverse, in a Banach spaceX are considered as aC0-group {exp(itlogA);tR} 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) foruD(A). Then ω±ilog(1+A) ism-accretive inX andilog(1+A) is the generator of aC0-group {(1+A)it;tR} of bounded imaginary powers, satisfying the estimate ‖(1+A)it‖ ≤ exp(ω|t|),tR. 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 {Ait;tR} forms aC0-group onX, with the estimate ‖Ait‖ ≤ exp(ω|t|),tR. This yields a slight improvement of the Heinz-Kato inequality.

MSC 1991

Primary 47B44 Secondary 47D03 


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© Birkhäuser Verlag 2000

Authors and Affiliations

  • Noboru Okazawa
    • 1
  1. 1.Department of MathematicsScience University of TokyoTokyoJapan

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