Free holomorphic functions on the unit ball ofB(H)n, II

Abstract

In this paper we continue the study of free holomorphic functions on the noncommutative ball B(H) n 1 := (X 1 , . . . , X n ) ∈ B(H) n : X 1 X * 1 + · · · + X n X * n 1/2 < 1 ,where B(H) is the algebra of all bounded linear operators on a Hilbert space H, and n = 1, 2, . . . or n = ∞. Several classical results from complex analysis have free analogues in our noncommutative setting. We prove a maximum principle, a Naimark type representation theorem, and a Vitali convergence theorem, for free holomorphic functions with operator-valued coefficients. We introduce the class of free holomorphic functions with the radial infimum property and study it in connection with factorizations and noncommutative generalizations of some classical inequalities obtained by Schwarz and Harnack. The Borel-Carathéodory theorem is extended to our noncommutative setting. Using a noncommutative generalization of Schwarz's lemma and basic facts concerning the free holomorphic automorphisms of the noncommutative ball [B(H) n ] 1 , we obtain an analogue of Julia's lemma for free holomorphic func-We also obtain Pick-Julia theorems for free holomorphic functions with operator-valued coefficients. We provide a noncommutative generalization of a classical inequality due to Lindelöf, which turns out to be sharper then the noncommutative von Neumann inequality. Finally, we introduce a pseudohyperbolic metric on [B(H) n ] 1 which is invariant under the action of the free holomorphic automorphism group of [B(H) n ] 1 and turns out to be a noncommutative extension of the pseudohyperbolic distance on B n , the open unit ball of C n . In this setting, we obtain a Schwarz-Pick type lemma. We also provide commutative versions of these results for operator-valued multipliers of the Drury-Arveson space.

A non-commutative F&M Riesz Theorem

“…The algebra L ∞ d can also be identified with the left multiplier algebra of H 2 d , viewed as a non-commutative reproducing kernel Hilbert space (RKHS) [26,16,1]. We remark that this left multiplier algebra is equal to the unital Banach algebra H ∞ d of all free NC functions in the NC unit row-ball B d N that are uniformly bounded in supremum norm [26,23]. The NC or free Toeplitz system is…”

Realizations of non‐commutative rational functions around a matrix centre, I: synthesis, minimal realizations and evaluation on stably finite algebras

“…The main underlying idea is that a function of d non-commuting variables is a function of d-tuples of square matrices of all sizes that respects direct sums and simultaneous similarities. See also the work of Helton-Klep-McCullough [38,39], of Popescu [61,62], of Muhly-Solel [57] and of Agler-McCarthy [1][2][3]. A crucial fact [48, is that nc functions admit power series expansions, called Taylor-Taylor series in honour of Brook Taylor and of Joseph L. Taylor, around an arbitrary matrix point in their domain.…”

Realizations of non-commutative rational functions around a matrix centre, I: synthesis, minimal realizations and evaluation on stably finite algebras

“…The main underlying idea is that a function of d noncommuting variables is a function of d−tuples of square matrices of all sizes that respects direct sums and simultaneous similarities. See also the work of Helton-Klep-McCullough [37,38], of Popescu [59,60], of Muhly-Solel [56], and of Agler-McCarthy [1,2,3]. A crucial fact [46, is that nc functions admit power series expansions, called Taylor-Taylor series in honor of Brook Taylor and of Joseph L. Taylor, around an arbitrary matrix point in their domain.…”