Spectral Radius Algebras and Shift

We consider spectral radius algebras associated to operators of the form h(S), where h ∈ H∞ and S is the unilateral shift. We show that, for a large class of H∞ functions, Bh(S) is weakly dense in LH. In this paper we continue the study of the spectral radius algebras (SRA) initiated in [2]. These algebras represent a very interesting class of non-selfadjoint, non-closed operator algebras. Furthermore, as we will see, the relationship between the operator to which an SRA is associated and the algebra itself is highly nonlinear. Finally, they are quite substantial, always containing the commutant of the operator in question. Let S be the unilateral shift acting on the Hilbert space H. One knows that, whenever h is a function in H∞, the Hardy space of essentially bounded functions on the disk D, one can define the operator h(S). The main object of the present study will be the spectral radius algebras associated with the latter operator. Before we continue we briefly introduce the relevant concepts and notation. Throughout the paper, H will be a complex, separable Hilbert space and it will be identified with the Hardy space H = H(D) in the usual manner. When A is an operator in L(H) (the algebra of all bounded linear operators on H) with spectral radius r = r(A) and m is a non-negative integer, we define the sequence of positive numbers dm = m/(1 + rm) and a sequence of positive, invertible operators Rm = (∑∞ m=0 d 2n m A ∗An )1/2 . The spectral radius algebra BA is the set {T ∈ L(H) : supm ‖RmTR m ‖ < ∞}. It is not hard to verify that BA is a unital algebra. The following result from [2] summarizes some of its important properties (cf. [2, Proposition 2.3, Corollary 2.4]). Proposition 1. Let A be an operator in L(H). Then T ∈ BA if and only if there exists M > 0 such that, for all x ∈ H and m ∈ N, ∑ n≥0 d 2n m ‖ATx‖ ≤ M ∑ n≥0 d 2n m ‖Ax‖. When AT = λTA, |λ| ≤ 1, and in particular if AT = TA, then T ∈ BA. It is easy to see that, when A is the unilateral shift S, every operator T satisfies the inequality in Proposition 1, so BS = L(H). In fact, more is true, as [1, Theorem 2.7] shows. Received by the editors February 12, 2009. 2000 Mathematics Subject Classification. Primary 47A65; Secondary 47B15, 47B20.