Remarks on a Hilbert Space of Analytic Functions.

ERGODIC THEOREMS* BY ALEXANDRA IONESCU TULCEA AND CASSIUS IONESCU TULCEA UNIVERSITY OF PENNSYLVANIA Communicated by Einar Hille, December 21, 1961 1. Let (Z. £, , be a complete totally a-finite measure space and E a Banach space. For each 1 < p < o denote by 4C. the vector space of all (Bochner) measurable mappings f of Z into E for which z oflf(z)|P is 4-integrable; here JE is endowed with the semi-norm f HfK||p = (Jzilf(z)jjPdAt(z))l/P Denote by LE the associated separated (Banach) space and by f -f the canonical mapping of 4e onto LE. Let SE be the vector space of all functions which are bounded and belong to £E; here SE is endowed with the semi-norm f ||If| = ess SupzIZ-Zf(z) j. Denote by SE the associated separated (normed) space and by f f the canonical mapping of SE onto SE. Let D be the set of all linear mappings T' of SE into SE such that' |TI1, < I and T1l'11. < 1. Then ||T||p < 1 for all 1 < p < o; hence, T can be extended by continuity to L' (we denote the extension by the same letter). For T E D and f ' o = U1 Up<=, 2P we denote by Tf a (determined) representative of the class Tf. 2. Let To, T1, . .. , TkE DU; consider the conditions: (1) To = I; (2) TjTj = TjTj for i,ji {O. 1,.. ., 1k; (3) TjTj+1 = Tj+1 for j E {O, 1, . . . , k 1}. We define To = I for allj E {O, 1, . . .,k, . For each functionf E V and each a > O let Gf(a) = {z |||f(z)H> a}. THEOREM 1. Let To, T1,.. ., Tk E D be k + 1 operators satisfying the conditions (1), (2), (3). For eachf E a and each a > 0, define G*(a) = Iz Supr oiI0,1. k), N II(Tq + Tj ± ... +Tj)f(z)/(n + 1)|I> a}. Then, for each set F E g verifying (except for sets of measure zero) the relations Gf(a) c F c Gf(a) we have aM(F) <. Fpjf(z)jld1A(z) < c2 COROLLARY 1. Let T E D. For eachf E VU and each a > 0, define E;(a) = {zJ Sup8ENJI(TO + T1 + ... + T')f(z)/(n + 1) || > al. Then, a/I(Ef(a)) < . Ef (a) If(Z) IdA(Z) < CCorollary 1 follows from Theorem 1 if we take k = 1, To = I, T. = T and F = E; (a).