THE （u＋К）-ORBIT OF ESSENTIALLY NORMAL OPERATORS AND COMPACT PERTURBATIONS OF STRONGLY IRREDUCIBLE OPERATORS

Let Н be a complex,separable,infinite dimensional Hilbert space,T∈L(Н),(U+κ)(T) denotes the (U+κ)-orbit of T,i.e.,(U+κ)(T)={R^-1 TR:R is invertible and of the form unitary plus compact}.Let Ω be an analytic and simply connected Cauchy domain in C and n∈N,A（Ω，n）denotes the class of operators,each of which satisfies (i) T is essentially normal;(ii)σ(T)=Ω^-,ρF（T）∩σ（T）=Ω；（iii）ind(λ-T)=-n,nul(λ-T)=0,(λ∈Ω）。it is proved that given T1,T2∈A(Ω，n）and c>0,there exists a compact operator K with ||K||<ε such that T1+K∈(u+κ)(T2),this result generalizes a result of P.S.Guinand and L.Marcoux6,15],Furthermore,the authors give a character of the norm closure of (u+κ)(T),and prove that for each T∈А(Ω，n）,there exists a compact (SI) perturbation of T whose norm can be arbitrarily small.

THE (U + K)-ORBIT OF ESSENTIALLY NORMAL OPERATORS AND COMPACT PERTURBATIONS OF STRONGLY IRREDUCIBLE OPERATORS

Let H be a complex, separable, infinite dimensional Hilbert space, T ∈ (H). (u+К)(T)denotes the (u+К)-orbit of T, i.e., (u +К)(T) = {R-1TR: R is invertible and of the form unitary plus compact}. Let Ω be an analytic and simply connected Cauchy domain in C and n ∈ N. A(Ω, n) denotes the class of operators, each of which satisfies (i) T is essentially normal; (ii) σ(T) = -Ω, pF(T) ∩σ(T)=Ω; (iii) ind (λ-T) = -n, nul(λ-T)＝0(λ∈Ω). It is proved that given T1, T2∈A(Ω, n) and e＞0, there exists a compact operator K with‖K‖＜e such that T1+K∈(u+К)(T2). This result generalizes a result of P. S. Guinand and L. Marcoux6,15]. Furthermore, the authors give a character of the norm closure of (u+К)(T),and prove that for each T ∈ A(Ω, n), there exists a compact (SI) perturbation of T whose norm can be arbitrarily small.