Rotaβs theorem for general functional Hilbert spaces

Author:
Joseph A. Ball

Journal:
Proc. Amer. Math. Soc. **64** (1977), 55-61

MSC:
Primary 47A45; Secondary 47A25

DOI:
https://doi.org/10.1090/S0002-9939-1977-0461176-0

MathSciNet review:
0461176

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Abstract | References | Similar Articles | Additional Information

Abstract: By a theorem of G.-C. Rota, every (linear) operator *T* on a Hilbert space with spectral radius less than one is similar to the adjoint of the unilateral shift *S* of infinite multiplicity restricted to an invariant subspace. This theorem is shown to be true in a rather general context, where *S* is multiplication by *z* on a Hilbert space of functions analytic on an open subset *D* of the complex plane, and *T* is an operator with spectrum contained in *D*. A several-variable version for an *N*-tuple of commuting operators with a corollary concerning complete spectral sets is also presented.

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Keywords:
Reproducing kernel function,
Riesz-Dunford functional calculus,
similarity,
complete spectral set and normal dilation

Article copyright:
© Copyright 1977
American Mathematical Society