On closures of joint similarity orbits |
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Authors: | Raúl E Curto Domingo A Herrero |
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Institution: | (1) Department of Mathematics, The University of Iowa, 52242 Iowa City, IA;(2) Department of Mathematics, Arizona State University, 85287 Tempe, AZ |
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Abstract: | For an n-tuple T=(T1,..., Tn) of operators on a Hilbert spacexxHx, the joint similarity orbit of T isxxSx(T)={VTV–1 =(VT1V–1,...,VTnV–1): V is invertible onxxHx}. We study the structure of the norm closure ofxxSx, both in the case when T is commutative and when it is not. We first develop a Rota-model for the Taylor spectrum and use it to study n-tuples with totally disconnected Taylor spectrum, in particular quasinilpotent ones. We then consider limits of nilpotent n-tuples, and of normal n-tuples. For noncommuting n-tuples, we present a number of surprising facts relating the closure ofxxSx(T) to the Harte spectrum of T and the lack of commutativity of T. We show that a continuous function which is constant onxxSx(T) for all T must be constant. We conclude the paper with a detailed study of closed similarity orbits.Research partially supported by grants from the National Science Foundation. |
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Keywords: | 47A10 47A15 47A53 47A55 47A65 47B38 |
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