The closures of (U + K)-orbits of essentially normal triangular operator models

Abstract

The (U + K)-orbit of a bounded linear operator T acting on a Hilbert space H is defined as (U + K)(T)=⨑ub;R ⁻¹ TR: R is invertible of the form unitary plus compact on H⫂ub;. In this paper, we first characterize the closure of the (U + K)-orbit of an essentially normal triangular operator T satisfying H = ∨⨑ub;ker(T − λI): λ ∈ ρ _F (T)⫂ub; and σ _p (T*) = Ø. After that, we establish certain essentially normal triangular operator models with the form of the direct sums of triangular operators, adjoint of triangular operators and normal operators, show that such operator models generate the same closed (U + K)-orbit if they have the same spectral picture, and describe the closures of the (U + K)-orbits of these operator models. These generalize some known results on the closures of (U + K)-orbits of essentially normal operators, and provide more positive cases to an open conjecture raised by Marcoux as Question 2 in his article “A survey of (U + K)-orbits”.