Composition operators on noncommutative Hardy spaces ✩

In this paper we initiate the study of composition operators on the noncommutative Hardy space H2 ball, which is the Hilbert space of all free holomorphic functions of the form f (X1, . . . , Xn) = ∞ k=0 |α|=k aαXα, α∈F + n |aα|2 < 1, where the convergence is in the operator norm topology for all (X1, . . . , Xn) in the noncommutative operatorial ball [B(H)n]1 and B(H) is the algebra of all bounded linear operators on a Hilbert space H. When the symbol ϕ is a free holomorphic self-map of [B(H)n]1, we show that the composition operator Cϕf := f ◦ ϕ, f ∈ H2 ball, is bounded on H2 ball. Several classical results about composition operators (boundedness, norm estimates, spectral properties, compactness, similarity) have free analogues in our noncommutative multivariable setting. The most prominent feature of this paper is the interaction between the noncommutative analytic function theory in the unit ball of B(H)n, the operator algebras generated by the left creation operators on the full Fock space with n generators, and the classical complex function theory in the unit ball of Cn. In a more general setting, we establish basic properties concerning the composition operators acting on Fock spaces associated with noncommutative varieties VP0 (H) ⊆ [B(H)n]1 generated by sets P0 of noncommutative polynomials in n indeterminates such that p(0) = 0, p ∈ P0. In particular, when P0 consists of thecommutators XiXj − XjXi for i, j = 1, . . . , n, we show that many of our results have commutative counterparts for composition operators on the symmetric Fock space and, consequently, on spaces of analytic functions in the unit ball of Cn. © 2010 Elsevier Inc. All rights reserved