Rota’s theorem for general functional Hilbert spaces

Abstract

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

Locations

• Proceedings of the American Mathematical Society - View - PDF