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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

On a functional calculus for decomposable operators and applications to normal, operator-valued functions


Author: Frank Gilfeather
Journal: Trans. Amer. Math. Soc. 176 (1973), 369-383
MSC: Primary 47A60; Secondary 47B15
MathSciNet review: 0312301
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Abstract: Whenever $ A = {\smallint _\Lambda } \oplus A(\lambda )\mu (d\lambda )$ is a decomposable operator on a direct integral $ H = {\smallint _\Lambda } \oplus H(\lambda )\mu (d\lambda )$ of Hilbert spaces and f is a function analytic on a neighborhood of $ \sigma (A)$, then we obtain that $ f(A(\lambda ))$ is defined almost everywhere and $ f(A)(\lambda ) = f(A(\lambda ))$ almost everywhere. This relationship is used to study operators A, on a separable Hilbert space, for which some analytic function A is a normal operator. Two main results are obtained. Let f be an analytic function on a neighborhood of the spectrum of an operator A. If $ f''(z) \ne 0$ for all z in the spectrum of A and if $ f(A)$ is a normal operator, then A is similar to a binormal operator. It is known that a binormal operator is unitarily equivalent to the direct sum of a normal and a two by two matrix of commuting normal operators. As above if $ f(A)$ is normal and in addition, $ f(z) - {\zeta _0}$ has at most two roots counted to their multiplicity for each $ {\zeta _0}$ in the spectrum of N, then A is a binormal operator.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0312301-4
PII: S 0002-9947(1973)0312301-4
Keywords: Normal operator, binormal operator, $ {W^\ast}$ algebra, direct integral reduction of a $ {W^\ast}$ algebra, functional calculus for operators
Article copyright: © Copyright 1973 American Mathematical Society