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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Rota's 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: 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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DOI: https://doi.org/10.1090/S0002-9939-1977-0461176-0
Keywords: Reproducing kernel function, Riesz-Dunford functional calculus, similarity, complete spectral set and normal dilation
Article copyright: © Copyright 1977 American Mathematical Society