ON THE POINT SPECTRUM OF NONSELFADJOINT PERTURBED OPERATORS OF WIENER-HOPF TYPE Ig. Cialenco

In this paper there are obtained results on the finiteness of the point spectrum of some nonselfadjoint operators. In particular the operators of WienerHopf type acting in arbitrary Hilbert space, l2 and L2(R+) are considered. In the present paper there is examined the problem of finiteness of the point spectrum of some nonselfadjoint operators. The similar problems have been studied for differential operators of second order [1,2] and fourth one [3], operator with finite difference of second order [4], Wiener-Hopf integral operator [5], for Friedrichs model [6] etc. Usually, in the case of nonselfadjoint perturbation, the problem of finiteness of the point spectrum is reduced to the theorem of uniqueness of analytical function. In this paper the problem is solved by the method of holomorphic extension of resolvent of unperturbed operator through continuous spectrum and application of the theorem of holomorphic operator-valued function. The obtained results are in concordance with those established in [1–6]. Moreover, it is possible to generalize substantially the results from [4], where it is considered the operator L generated by the difference expression (Ly)j = 1 2 (yj−1 + yj+1) + bjyj , (j = 1, 2, . . . ) where (yj) ∈ l2; y0 = θy1; bj ∈ C (j = 1, 2, . . . ); θ ∈ C. There is considered the operator with finite difference of any order and more general perturbation (not necessarily diagonal). In the first section, we give a general scheme about finiteness of the point spectrum of perturbed operators. In section 2, we prove an abstract theorem for the case of abstract Wiener-Hopf type operators. The sections 3–5 contain various applications of the abstract theorem. Thus, it is considered the operator generated by generalized Jacobi matrix, the case when unperturbed operator is with finite difference (arbitrary order) acting in l2 and L2(R+) respectively. The results of the present paper generalize our previous ones given in [7–8]. 2. Throughout the paper, H will denote a Hilbert space, B(H) the class of all linear and bounded operators on H, B∞(H) the class of compact operators on H. Let H0 and B be linear and bounded operators on H such that the following assumptions are fulfilled: (1) The operator H0 is selfadjoint and σp(H0) = ∅; ANALELE ŞTIINŢIFICE ALE UNIVERSITǍŢII ”AL.I.CUZA” IAŞI, t.XLIV, s.I.a, Matematica, 1998, Supliment