Composition operators on noncommutative Hardy spaces

In this paper we initiate the study of composition operators on the noncommutative Hardy space Hball2, which is the Hilbert space of all free holomorphic functions of the formf(X1,…,Xn)=∑k=0∞∑|α|=kaαXα,∑α∈Fn+|aα|2<1, where the convergence is in the operator norm topology for all (X1,…,Xn)(X1,…,Xn) in the noncommutative operatorial ball [Bn(H)]1[B(H)n]1 and B(H)B(H) is the algebra of all bounded linear operators on a Hilbert space HH. When the symbol φ   is a free holomorphic self-map of [Bn(H)]1[B(H)n]1, we show that the composition operatorCφf:=f∘φ,f∈Hball2, is bounded on Hball2. 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 Bn(H)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 CnCn. In a more general setting, we establish basic properties concerning the composition operators acting on Fock spaces associated with noncommutative varieties VP0(H)⊆[Bn(H)]1VP0(H)⊆[B(H)n]1 generated by sets P0P0 of noncommutative polynomials in n   indeterminates such that p(0)=0p(0)=0, p∈P0p∈P0. In particular, when P0P0 consists of the commutators XiXj−XjXiXiXj−XjXi for i,j=1,…,ni,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 CnCn.