Invariant Subspaces for Algebras of Subnormal Operators. Ii

We continue our study of hyperinvariant subspaces for rationally cyclic subnormal operators. We establish a connection between hyperinvariant subspaces and weak-star continuous point evaluations on the commutant. Introduction. Let A be a compact subset of the complex plane C and let R(K) denote the algebra of rational functions with poles off K. For a positive measure p with support in K let R2(K,p) denote the closure of R(K) in L2(p). Every rationally cyclic subnormal operator is unitarily equivalent to multiplication by z, Mz, on an R2(K, p)-space [1, p. 146]. Under this representation each operator that commutes with Mz is represented by multiplication by a function in R2 (K, p) n L°°(p), and conversely [1, p. 147]. In [2] we proved the existence of invariant subspaces for the algebra R2 (K, p) n L°°(p). In this paper we show there exist weak-star continuous point evaluations on R2(K,p) n L°°(p) if L2(p) ^ R2(K,p), and we show that these point evaluations give rise to the kinds of invariant subspaces found in [2]. Finally, we establish a connection between analytic bounded point evaluations on iü2-spaces and ñp-spaces for 2 < p < 4. Notation. Let m denote area measure on C. For a complex measure u with compact support in C let v(z) = \(c — z)~l dv(<¡) and v(z) = / \ç — z\~l d\v\(ç). By Tonelli's theorem v(z) < oo m-a.e. and hence the Cauchy transform ù is defined m-a.e. For g in ¿HM) let 9 equal (gdv)~. THEOREM. Let g G Ä2(A,/i)x and let A = {z: g(z) £ 0}. Then for m-a.e. z in A there exists a weak-star continuous multiplicative linear functional ez on R2(K,p) nL°°(p) such that ez(f) = f(z) for each f in R(K). Moreover, for ma.e. such z there exist x and y in R2(K,p) such that ez(f) = (fx,y) for each f in R2(K,p)nL°°(p). REMARK. Let z in A be such that both conclusions of the theorem hold, and let x and y be as in the second conclusion. Let H be the closed linear span of {(c z)fx: f G R2(K,p) n L°°(p)} in L2(p). Since y _L H, it follows that H is a nontrivial hyperinvariant subspace for Mc on R2(K,p). LEMMA 1. Let p G (2,4) and let s = 2/(p 2). Suppose fn G L2(p) and ||/n||2 < 2~n for each positive integer n. Then there exists a function w : C —► (0,1] such that w_1 G Ls(p), fn G Lp(wdp) for each n, and fn —+ 0 in Lp(wdp). Received by the editors August 13, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 47B20.