Between the unitary and similarity orbits of normal operators

Abstract

has defined the (^ 4-Jf)-orbit of an operator T acting on a Hubert space iF to be (^ + Jr)(Γ) = {R~ιTR: R invertible of the form unitary plus compact}. In this paper, we characterize the norm closure in £ §(%?) of such an orbit in three cases: firstly, when T is normal; secondly when T is compact; and thirdly, when T is the unilateral shift. Some consequences of these characterizations are also explored.1. Introduction. Let %f be a complex, separable, infinite dimensional Hubert space and denote by 3 §(%?) the set of bounded linear operators acting on %?. As usual, 3£{%?) will denote the unique twosided ideal of compact operators. There are many interesting ways of partitioning the set 38{%?) into equivalence classes. We mention two in particular.Given T e 33{%T), we define the unitary orbit of T as W{T) = {U*TU: U e 33{%?) a unitary operator}. Then an operator A e %(T) if its action on %? is geometrically identical to that of T. Equivalently, one can think of A as T itself acting on an isomorphic copy of %r.Another much studied class is the similarity orbit of T e 3S[^), namely S?{T) = {S~ιTS: S e 33{fT) an invertible operator}. This notion of equivalence ignores the geometry of the Hubert space, and concentrates on the underlying vector space structure.In general, neither of these sets need be closed. This is in contrast to finite dimensional Hubert spaces, where %/(T) is always closed while S*(T) is closed if and only if T is similar to a normal matrix [Her 1, p. 14]. It is therefore interesting to describe the norm closure of these orbits, a program which for unitary orbits was undertaken by D. W. Hadwin One can also turn one's attention to the Calkin algebra s/ffi) = £%(^)/J?(β?) and consider both unitary and similarity orbits there. Indeed, one of the major results along these lines is the classification

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

“…Although Marcoux's conjecture remains open, the results in this paper provide more positive cases to it, and generalize some results on (U + K)-orbits investigated in [7][8][9][10][17][18][19][20][21]. Concretely, in Section 3, we characterize the closures of the (U + K)-orbits of certain essentially normal triangular operators.…”

“…In 1986, Herrero [13] raised a problem which was that given T ∈ L(H), to find a simple characterization of the closure of (U + K)(T ). This problem was subsequently studied for various classes of essentially normal operators (see, for example [1,7,9,10,19], etc.). In 1992, Marcoux [19] formulated the conjecture that (U + K)(R) = (U + K)(T ) whenever R and T are essentially normal and have the same spectral picture, and Marcoux [19,Corollary 2.6], Jiang and Wang [17,Theorem 5.5] provided two positive cases to this conjecture.…”

Strongly approximative similarity of operators

“…The notion of (I + K)-orbit was introduced by P. S. Guinand and L. Marcoux [7], and the latter has studied it in several subsequent papers.…”

Similarity Invariants of Essentially normal Cowen-Douglas Operators and Chern Polynomials

“…Herrero [27] ask if we can find a simple characterization of the closure of (U + K)-orbit of a Hilbert space operator. In 1993, Guinand and Marcoux [23,24] completely characterized the the closure of (U + K)-orbits of normal operators, compact operators and certain weighted shifts respectively. In 1998, the first author and Wang [42] completely characterize the closure of (U + K)orbits of certain essentially normal operators.…”