It is shown that the finiteness of eigenvalues in a spectral gap of a definitizable or locally definitizable selfadjoint operator in a Krein space is preserved under finite rank perturbations. This results is applied to a class of singular Sturm-Liouville operators with an indefinite weight function.

Abstract. The spectral properties of a singular left-definite Sturm-Liouville operator JA are investigated and described via the properties of the corresponding right-definite selfadjoint counterpart A which is obtained by substituting the indefinite weight function by its absolute value. The spectrum of the J -selfadjoint operator JA is real and it follows that an interval

Let A and B be selfadjoint operators in a Krein space and assume that the resolvent difference of A and B is of rank one. In the case that A is nonnegative and I is an open interval such that σ(A) ∩ I consists of isolated eigenvalues we prove sharp estimates on the number and multiplicities of eigenvalues of B in I. The general result is illustrated with eigenvalue estimates for singular indefinite Sturm-Liouville problems.

Singular Sturm-Liouville operators with the indefinite weight sgn(·) and a symmetric potential which has a positive limit at ∞ have a gap in the essential spectrum. Under an additional condition it is shown that in this gap are no eigenvalues.

In this note we give sufficient conditions for the coefficients of a singular Sturm-Liouville operator on R with an indefinite weight function such that the operator is similar to a self-adjoint operator in the Hilbert space L 2 (R). We consider the the maximal Sturm-Liouville operator A associated to the differential expressionloc (R) are real-valued and p > 0 a.e., and AC(R) denotes the set of absolutely continuous functions on R. We assume that τ is in limit point case at ±∞, hence A = A * , and also that A is uniformly positive, i.e. (Af, f ) ≥ 0, f ∈ domA and 0 ∈ ρ(A). We denote by J the multiplication operator in L 2 (R) with the sign function sgn and consider the operator JAThe indefinite Sturm-Liouville operator JA is not self-adjoint in the Hilbert space L 2 (R). We give a criterion for JA to be similar to a self-adjoint operator, i.e. there exists a bijection Theorem 1 If the uniformly positive operator A has the same form domain as A − q and the functions p and 1 p are essentially bounded in a neighborhood of 0, then JA is similar to a self-adjoint operator. P r o o f. We prove this theorem in two steps. 1. We show that the form domainWe assume without loss of generality that q ≡ 0 since A has by assumption the same form domain as A − q. Consider the operator P with domP = D, P f = √ pf . This operator is a closed operator in L 2