The closures of \((\mathcal U + \mathcal K)\)-orbits of essentially normal triangular operator models
The \((U + K)\)-orbit of a bounded linear operator \(T\in B(H)\) is defined as the set \((U + K)(T)\) of all \(R^{-1}TR\) such that \(R\) is invertible of the form of a unitary plus a compact on \(H\). Regarding a problem posed by \textit{D.\,A.\thinspace Herrero} [Am.\ J.\ Math.\ 108, 451--484 (1986; Zbl 0592.47011)], the authors characterize the closure of the \((U + K)\)-orbit of an essentially normal triangular operator \(T\) satisfying \(H=\bigvee\{\ker(T-\lambda I): \lambda \in \rho_F(T)\}\) and \(\sigma_p(T^*)=\emptyset\). They establish certain essentially normal triangular operator models, which are analogous to the direct sum of the diagonal entries of Apostol-Morrel simple models introduced in [\textit{C.\,Apostol, L.\,A.\thinspace Fialkow, D.\,A.\thinspace Herrero} and \textit{D.\,Voiculescu}, ``Approximation of Hilbert space operators.\ II'' (Research Notes in Mathematics 102; Boston--London--Melbourne: Pitman) (1984; Zbl 0572.47001)]. They then show that such operator models generate the same closed \((U + K)\)-orbit if they have the same spectral picture. \textit{L.\,W.\thinspace Marcoux} [Indiana Univ.\ Math.\ J.\ 41, No.\,4, 1211--1223 (1992; Zbl 0767.47006)] formulated the conjecture that \(\overline{(U + K)(R)} = \overline{(U + K)(T )}\) whenever \(R\) and \(T\) are essentially normal and have the same spectral picture [see \textit{L.\,W.\thinspace Marcoux}, in: Operator theory and Banach algebras.\ Proceedings of the international conference in analysis, Rabat, Morocco, April 12--14, 1999, 91--114 (2003; Zbl 1084.47501)]. The authors treat this conjecture as well.
scientific article; zbMATH DE number 5723659