The present note is devoted to the study of spectral properties of the operator function L(A) = A(A)-I,where A(A) is a continuous self-adjoint compact operator function in a Hilbert space 7-/with inner product (*, ,) and I is the identity operator on ~. Moreover, we assume that the following condition holds:for any nonzero vector u E "H, the functionincreases in a neighborhood of any of its zeros (if they exist). Clearly, under this condition the function f~ can have at most one zero (possibly multiple).The spectral properties of self-adjoint operator functions with property (2) were studied by means of the Rayleigh functional in [1-3]. However, we note that in these papers the Rayleigh functional was defined under the following assumptions: first, for any nonzero u E 7"i, the function f~ has a zero p(u) e (a, b); second, either the operator function L(A) is continuously differentiable and the zero p(u) of the functionThe motivation for studying operator functions of the form (1)-(2) is explained in [4,5], in which the asymptotic behavior as e --. 0 of the spectrum of the Neumann problem in thick singularly degenerate unions depending on a small parameter ~ was investigated. In these papers it was shown that the limit (~ = 0) spectral problem is equivalent to the spectral problem for the operator function S(A)=(A2-F1)A § AE(Trn-r/2,~rn+r/2), hEN, where A and B are self-adjoint compact operators on ~, A > 0, and B > 0. We can also reduce the quadratic operator pencil T(A) = A21 + A(aA + B) + A + C, considered in [6], to an operator function of the form (1); here A = A* >> 0, the inverse operator A-' is compact, and the operators B and C are symmetric and A is compact. The spectra of the pencils T(,~) and TI(A) = -(I + coincide, and the pencil TI(A) on the interval (a3,-l/a) satisfies conditions (1)-(2).We can readily see that the methods of [1-3] cannot be applied to the operator functions S(A) and T(A), and the method suggested in [4,5,7] is not general for the operator functions of the form (1)-(2).In the present note we construct a singular Rayleigh functional and apply it to obtain the following results:1) the spectrum of the operator function L(A) on (a, b) is discrete; 2) the only possible accumulation point of the eigenvalues is the point A --b; we obtain a criterion for accumulation; 3) variational principles are proved for the eigenvalues.
