Complex Geometry and Operator Theory
نویسندگان
چکیده
One of the principal goals of spectral theory for operators is to find unitary invariants which are local relative to the spectrum. Multiplicity theory provides a complete set of such invariants for normal operators on (complex) Hilbert space. For general operators on finite-dimensional Hilbert space a nilpotent operator is attached to each point of the spectrum and these "local operators" together with their relative location provide a complete set of unitary invariants. In this note we announce analogous results for a class of operators whose characteristic property is having an open set of eigenvalues. Included in this class are the backward shift together with the adjoint of various subnormal, hyponormal, and weighted shift operators. Although our goal is to provide a systematic spectral theoretic approach to the study of this type of operator, here we are concerned only with a result on unitary equivalence. For 12 a connected open subset of C and n a positive integer, let B„(12) denote the (bounded linear) operators T defined on the separable Hilbert space H which satisfy: (1) 12 is contained in the spectrum o(T); (2) ( r co)ff = H for co in 12; (3) dim ker(r co) = n for co in 12; and (4) Vwe«ker(!Tco) = H. To study T in Bw(12) we introduce the "local operators" N^ on Ww defined for each co in 12, where Ww = ker(!Tco) fI + 1 and N^ = (2Tco)|Nw. Observe that Nu is nilpotent of order n + 1 on a space of dimension n(n 41). Our principal result is
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