Invariant subspaces for algebras of subnormal operators. II

Every rationally cyclic subnormal operator has a hyperinvariant subspace. A bounded linear operator on a Hilbert space is defined to be subnormal if it is the restriction to an invariant subspace of a bounded normal operator. S. W. Brown [4] proved that every subnormal operator has a nontrivial invariant subspace. An idea in his proof leads to a stronger theorem, whose proof is easier. THEOREM. Let jt be a compactly supported positive measure on the complex plane C. Let H be a closed subspace of L2(Mu) with 1 E H. Let A be a subalgebra of LX(y) containing the function z such that AH C H. Then there exists a nontrivial closed subspace K of H such that AK c K. Before proving the theorem, we will point out some consequences. A subnormal operator S on a space H is rationally cyclic if there exists a vector x in H such that the set {r(S)x: r e Rat(a(S))} is dense in H, where Rat(a(S)) denotes the algebra of rational functions with poles off the spectrum of S. For each such S, there exists a measure jt such that S is unitarily equivalent to multiplication by z on R2(u(S), M), the closure in L2(M) of Rat(u(S)) [5, p. 146]. Under this representation each operator that commutes with S is represented by multiplication by a function in R2(u(S), jt) n LX(y), and conversely [5, p. 147]. A subspace invariant for every operator that commutes with S is called hyperinvariant. If we let H = R2(u(S), jt) and A = R2(u(S), jt) n LX(y) in the theorem, then we obtain a hyperinvariant subspace. COROLLARY 1. Every rationally cyclic subnormal operator has a hyperinvariant subspace. The following is a trivial consequence of Corollary 1. COROLLARY 2. Every subnormal operator S has a subspace invariant for the algebra {r(S): r E Rat(a(S))}. Received by the editors April 7, 1985. 1980 Mathematics Subject Classification. Primary 47B20. Kei 'ords atSnd phrases. Subnormal operators, invariant subspaces. 1 Research supported in part by a grant from the National Science Foundation. '(1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page 462 This content downloaded from 157.55.39.44 on Sat, 24 Sep 2016 05:37:28 UTC All use subject to http://about.jstor.org/terms ALGEBRAS OF SUBNORMAL OPERATORS 463 An excellent general reference for subnormal operators is the book by J. B. Conway [5]. J. E. Brennan [1-3] has worked extensively on rationally invariant subspaces, and the author has found that work instructive. PROOF OF THE THEOREM. If H = L2(ML), then let K = fH, where f is a nontrivial characteristic function. Assume H * L2(M). Then there exists a nonzero function g in L2(M) such that ffg d,u = 0 for every f in H. Let p = 3 and q = 3/2 for the remainder of this paper. By [1] and [5, p. 316] there exists a point w in C such that g(z)(z w)-1 is in Ll([), 4({ w }) = O and f g(z)(z w)1dM(z) * 0. Fix such a point w. Let AP(M) denote the closure in LP(M) of A. Note that AP(M) C H. Define a bounded linear functional L on AP(M) by L(f) = f (z)g(z)(z w)1dM(z) for f in A P(M). By the Hahn-Banach theorem there exists a norm-preserving extension of L to LP(M). This extension is represented by a function h in Lq([) with llhllq = IILiI. That is,