Rota’s theorem for general functional Hilbert spaces

Ball, Joseph A.

doi:10.1090/S0002-9939-1977-0461176-0

Rota’s theorem for general functional Hilbert spaces
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by Joseph A. Ball

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

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

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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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