 
Summary: FINITEDIMENSIONAL PERTURBATIONS
OF SELFADJOINT OPERATORS
Jonathan Arazy and Leonid Zelenko
Abstract.
We study finitedimensional perturbations A + B of a selfadjoint operator A acting in
a Hilbert space H. We obtain asymptotic estimates of eigenvalues of the operator A + B
in a gap of the spectrum of the operator A as 0, and asymptotic estimates of their
number in that gap. The results are formulated in terms of new notions of characteristic
branches of A with respect to a finitedimensional subspace of H on a gap of the spectrum
(A) and asymptotic multiplicities of endpoints of that gap with respect to this subspace. It
turns out that if A has simple spectrum then under some mild conditions these asymptotic
multiplicities are not bigger than one. We apply our results to the operator (Af)(t) = tf(t)
on L2([0, 1], c), where c is the Cantor measure, and obtain the precise description of the
asymptotic behavior of the eigenvalues of A + B in the gaps of (A) = C(= the Cantor set).
1. Introduction
An extensive literature exists on the perturbations of selfadjoint operators. In many
cases ones studies perturbations of the form:
(1.1) A() = A + B,
where A, B are selfadjoint operators, acting on a Hilbert space H, is a real or com
plex parameter. If 0 is an isolated eigenvalue of A, then it is possible to find branches
