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

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- by Frank Gilfeather PDF
- Trans. Amer. Math. Soc.
**176**(1973), 369-383 Request permission

## 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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## Additional Information

- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**176**(1973), 369-383 - MSC: Primary 47A60; Secondary 47B15
- DOI: https://doi.org/10.1090/S0002-9947-1973-0312301-4
- MathSciNet review: 0312301