then Theorem 5 is a refinement in our simple situation of a deep result of Petras [6], prov- ing a general theorem on the decay of a series of resonances. Our formulas allow us not only to solve in principle the question on the decay of the sets of resonances in series but also to compute asymptotically the resonances under an 'almost closed trap," l~I>>l, in particular, to determine their lifetime 1. 2. 3. 4. 5. 6. LITERATURE CITED D. N. Clark, "One-dimensional perturbation of restricted shifts," J. Anal. Math., 25, 169-191 (1972). B. S. Pavlov and M. D. Faddeev, "Spectral analysis of unitary perturbations of contrac- tions," Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst., 115, 215-227 (1982). N. N. Voitovich, B. Z. Katsenelenbaum, and A. N. Sivov, The Generalized Method of Eigen- oscillations in Diffraction Theory [in Russian], Nauka, Moscow (1977). N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Vol. II, Ungar, New York (1963). P. D. Lax and R. S. Phillips, Scattering Theory, Academic Press, New York (1967). S. V. Petras, "On the splitting of series of resonances on the 'nonphysical' sheet," Funkts. Anal. Prilozhen., 9, No. 2, 89-90 (1975). INVARIANT SUBSPACES OF TOEPLITZ OPERATORS V. V. Peller UDC 517.5 This paper is devoted to the problem of the existence of invariant subspaces for Toeplitz operators. Let F be a Lipschitzian arc in the plane and let f be a non- constant continuous functions on the unit circumference. It is proved that if there exists an open circle ~ such that ~(T)Nrfl~ ~ ~(~)0 (~\r)# ~ and if the modulus of continuity mf of the function f satisfies the condition 0 then the Toeplitz operator Tf in the Hardy space H 2 has a nontrivial hyperinvari- ant subspace. For the proof of this theorem one makes use of the Lyubich--Matsaev theorem. i. Introduction The fundamental purpose of this paper consists in the proof of the existence of nontri- vial hyperinvariant subspaces for a suff{ciently large class of Toeplitz operators, playing an important role in the theory of operators and in function theory and having many applica- tions in various domains (see [1-5]). Let f be a bounded function on the unit circumference ~ = {~:I~I =I } . The Toeplitz operator Tf, acting in the Hardy space H 2, is defined by the equality - eH Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matem~ticheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 170-179, 1983. 0090-4104/84/2701-2533508.50 9 1984 PlenumPublishing Corporation 2533