On closures of joint similarity orbits
For an n-tuple \(T=(T_ 1,...,T_ n)\) of operators on a Hilbert space \({\mathcal H}\), the joint similarity orbit is \[ {\mathcal S}(T)=\{VTV^{- 1}:=(VT_ 1V^{-1},...,VT_ nV^{-1}):\quad V\quad is\quad invertible\quad on\quad {\mathcal H}\}. \] The paper is devoted to a careful study of the structure of the norm closure of \({\mathcal S}(T)\) (in \({\mathcal L}({\mathcal H})^ n)\), both in the commutative and noncommutative cases. By developing a Rota model for commuting n-tuples over the Taylor spectrum, the authors study \(\overline{{\mathcal S}(T)}\) for T quasi-nilpotent or, more generally, with totally disconnected spectrum. They then consider limits of nilpotent and normal n-tuples. By means of a number of examples, it is seen how different the several variables theory is from that for a single operator. Moreover, the noncommuting case offers many surprising facts relating the closure of \({\mathcal S}(T)\) to the Harte spectrum of T and the lack of commutativity. The paper also deals with closed joint similarity orbits; in addition, it is shown that a continuous function on \({\mathcal L}({\mathcal H})^ n\) which is invariant under joint similarity must be constant.
Rota model for commuting n-tuples over the Taylor spectrum
scientific article; zbMATH DE number 3909142
scientific article