INVARIANT SUBSPACES OF THE OPERATOR OF MULTIPLICATION BY z IN THE SPACE EP IN A MULTIPLY CONNECTED DOMAIN D. V. Yakubovich UDC 517.984 In the paper one obtains the description of invariant subspaces of the multiplication operator ~(~I ~(~), in the Hardy--Smirnov space bY(G), where G is a finitely connected domain with a piecewise C 2- smooth boundary. For the case of an analytic "interior boundary" Pint of the domain G and p = 2, a more precise description is given, generalizing the Hitt--Sarason result on the invariant subspaces of the space H2 in a circular annulus. Let G be a bounded multiply connected domain in C with a boundary consisting of a finite number of closed Cl-smooth pairwise disjoint curves (such domains will be said to be admissible) and let 1 ___ p _ oo. In the paper we obtain the description of the (closed) subspaces ~ in EP(G), invariant with respect to the multiplication operator Mzx(Z) = zx(z). Here EP(G) is the Hardy--Smirnov space [1]; E~*(G) is considered with the *-weak topology. Earlier, this problem has been solved for the following special cases. Sarason in [2] has considered the case when G = {a < [ z [ < b} is a circular annulus and ~ is rationally invariant, i.e., for all rational functions Q with poles outside G. Hasumi, Voichick [3, 4] have obtained the description of rationally invariant subspaces for an arbitrary domain G. All the invariant subspaces in an annulus for the case p = 2 have been described recently by Hitt [5]. However, in a certain sense Hitt's result is not complete and this deficiency has been corrected by Sarason [6]. (below, in the appendix we present more direct proofs of certain lemmas from [5, 6]). The methods of Hitt and Sarason make use in an essential manner of Hilbert space techniques and of the fact that G is a circular annulus. For an arbitrary domain G with a smooth boundary and arbitrary p, Royden [7] has obtained the description of the invariant subspaces, subjected to a certain sufficiently restrictive hypothesis. The description, presented in this paper, of all invariant subspaces in EP(G) differs from the Hitt--Sarason description. If the boundary of the domain G contains "angles", then one detects certain effects that are absent in the cases considered by all the enumerated authors (see Remarks 2, 3 to Theorem 1). The obtained description, just as Hitt's one [5], is incomplete in the sense that the class of functional parameters is not characterized. For the case p = 2 and an analytic boundary of the domain G, we present another description, without this deficiency. The technique of the proof differs from [5, 6]; this paper can be considered as a continuation of [8, Chap. 4]. In the proof of Theorem 2 we apply the Hitt--Sarason theorem. In an essential manner we make use of B. M. Solomyak's [9, 10] method of analytic continuation and the "sewing" of Cauchy integrals. The paper answers a series of questions posed by Hitt in [5, Sec. 9]. Royden's conjecture [7, p. 152] remains open. 1. Necessary Definitions and Formulation of Results. We recall the concepts related to the classes EP and HP (see [11, 12, 7] etc.). Let f] be an admissible (simply or multiply connected) domain. A function 4) E H~ is said to be inner if I 4~ I = const a.e. on each component of the boundary 0fl (see [7]; in [12] one has given a somewhat different definition). Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 178, pp. 166-183, 1989. 2046 0090-4104/92/6006-2046512.50 Plenum Publishing Corporation