Logarithms and imaginary powers of closed linear operators |
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Authors: | Noboru Okazawa |
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Institution: | (1) Department of Mathematics, Science University of Tokyo, 162-8601 Tokyo, Japan |
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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 ), 0(/2), in a Hilbert spaceX. That is, |Im(Au, u)| (tan )Re(Au, u) foru D(A). Then ±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(|t|),t R. In particular, ifA is invertible, then ±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(|t|),t R. This yields a slight improvement of the Heinz-Kato inequality. |
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Keywords: | Primary 47B44 Secondary 47D03 |
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