N N On the other hand, for N < n= the inequality ll~.=, ~%11<i2(~__=,I~ I,):/' follows automatically from the property i). If, however, nio < N i nio+~, where io > i, then N ~ n~+l i~ ni-bl N N = £=: :-ni+i "= ! = ~=* ~e Lena is proved. ~EO~M~ In every inflnite-dlmensional Banach space there exist series with nonlinear domains of the sum. ~is theorem is a direct consequence of Le~as 1 and 2. . 2. 3. 40 LITERATURE CITED E. M. Nikishin, Mat° Sb., 85, No. 2, 272-285 (1971). P. A. Kornilov, Mat. Sb., ~3~3, No. 4, 958-616 (1980). V. Do Mil'man, Funkts. Anal. Prilozhen., ~, No. 4, 28-37 (1971). J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Vol. I, York (1977). SpringeruVerlag, New FINITENESS OF THE DISCRETE SPECTRUM OF THE PERTURBED WIENER-HOPF OPERATOR L. V. Kakabadze UDC 517.9 We consider the Weiner--Hopf operator (Af)(z)=~k(x--y)f(y)dy, x>0, k(x)~L~(R), (1) 0 which acts in the space L2(R+), where R+ is the positive semiaxis. The symbol of A is the Fourier transform of the kernel: ~ ~)= ~k (z)eiX~dx,% ~ R. It defines a continuous closed ~ R curve F= {~(%): %~R},0~F in the complex plane C. The set ~= {~C\F: ~nd(~--~(l))l~=_~= 0} is the resolvent set of A [i].~ When A is perturbed by means of a completely continuous operator T in ~, there arises a discrete set sd of eigenvalues of finite multiplicity. In this article, we obtain sufficient conditions under which the discrete spectrum ~d is finite. The proof is based on a study of the analytic properties of the Fredholm determinant of a certain operator-function ~(~) which is holomorphic in ~. Fredholm determinants play an important role in the study of the spect- ral properties of certain non-self-adjoint operators [2]. They have also been applied in the study of eigenvalues and resolvents of certain generalizations of Friedrichs models [3]. We note that sufficient conditions for the finiteness of the discrete spectrum of A + T, T = T*, where A is a self-adjoint Wiener-~opf operator (abstract, discrete, or integral), have been obtained by other methods in [4-6]. We introduce some notation~ C± = {z ~ C: ±im z > 0}, Cs~ = {z + is~ z ~ C±}, s ~ R~ II s = {z~ C~ lim z I < s}, s > 0. We will assume that, for some r > 0 the kernel k(x) satisfies the condition e~ -~ Ik(x)~ L: (R). (i) Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozhenlya, Vol. 20, No. 4, pp. 76-78, October-December, 1986. Original article submitted July 22, 1985. 0016-2663/86/2004-0319512.50 © 1987 Plenum Publishing Corporation 319