The aim of this paper is to find estimates of the Hausdorff distance between the spectra of two nonselfadjoint operators. The operators considered are assumed to have their imaginary parts in some normed ideal of compact operators. In the case of the classical Schatten ideals the estimates are given explicitly. It is well known that (σ(A), σ(B)), the Hausdorff distance between the spectra of two selfadjoint operators A, B, is bounded by A − B . The same estimate also holds for normal operators.

more »

... This is related to the fact that the norm of the resolvent of such an operator is equal to 1/d(λ, σ(A)). For nonselfadjoint operators A, B acting in an n-dimensional space an estimate for the distance between their spectra has been obtained by R. Bhatia and K. K. Mukherjea in [1], and L. Elsner The dependence of the bound on the nth root of A − B is related to the fact that the resolvent norm of an operator can behave in the neighborhood of an eigenvalue as (1/d(λ, σ(A))) n . Looking at the form of this estimate one sees that it cannot be generalized to compact operators acting in infinite-dimensional Hilbert space. The class of compact operators is too large. The situation changes if we restrict ourselves to normed ideals of compact operators. In [7, Theorem 3] it was shown that for each normed ideal (with norm not equivalent to the operator norm) there exists an estimate of this kind. In the case of the Schatten ideal S p , 1 ≤ p < ∞, there exists a constant C p such that 1991 Mathematics Subject Classification: 47A55, 47B10.