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The spectral theory of the infinitesimal generator of an arbitrary one-parameter group of isometries on Hp of the torus, 1 < p < oo, p # 2, is considered. In particular, the spectrum of the generator is determined. 0. Introduction. The space Hp (1 < p < +co) of the torus C2 is defined (see [8]) as the subspace of complex LP(C2) consisting of those functions whose double Fourier coefficients vanish on the complement of P, where P is the "positive set" in the character group of C2, P = {(m,n): « > 0} U {(m,0): m > 0}. Throughout what follows we shall use (z,w) (z = e'9,w = e'*) to denote the general point of C2, Zp denotes the closure in LP(C2) of the polynomials in z, and we identify Zp with Hp(d9/2ir). The strongly continuous one-parameter groups [Tt] of isometries on HP(C ), 1 < p < co,p # 2, were described in [3]. The present paper is devoted to the spectral analysis of their infinitesimal generators, and thereby completes the investigation initiated in [3]. It is known that {7^} leaves Zp invariant (see (1.1) below), and so the study of {Tt} and its infinitesimal generator A naturally splits into two cases according to whether [Tt\Zp) has a bounded generator (§1) or an unbounded generator (§2). In the former case there are a real constant p and a real-valued measurable function 8(f) on the circle C such that (0.1) (Ttf)(z,w) eip'f(z,eiS{z),w) for/ E HP(C2). As shown in §1, in this case the spectrum of A, A(A), is the closure of the set ;{p + L£L0« [ess. range (5)]}. The dependence on 8(f) of the fine structure of A(A) is examined in the process. When {Tt\Zp} has an unbounded generator, it is shown in [3, Theorems (2.22) and 2.24)] (by cocycle methods reminiscent of [6, Lecture V]) that there Received by the editors December 24,1975. AMS (MOS) subject classifications (1970). Primary 46E15, 46E30, 47A10, 47D05.

One-parameter groups of isometries on Hardy spaces of the torus: spectral theory