Free Pick functions: Representations, asymptotic behavior and matrix monotonicity in several noncommuting variables

Abstract

We extend the study of the Pick class, the set of complex analytic functions taking the upper half plane into itself, to the noncommutative setting. R. Nevanlinna showed that elements of the Pick class have certain integral representations which reflect their asymptotic behavior at infinity. Löwner connected the Pick class to matrix monotone functions. We generalize the Nevanlinna representation theorems and Löwner's theorem on matrix monotone functions to the free Pick class, the collection of functions that map tuples of matrices with positive imaginary part into the matrices with positive imaginary part which obey the free functional calculus. ContentsJ. E. PASCOE RYAN TULLY-DOYLE 3.1.1. The coefficient Hardy space of a general free vectorvalued function 23 3.2. Models 24 3.3. The lurking isometry argument for linear forms 24 4. Löwner's theorem 26 4.1. The Hamburger model 27 4.2. The Hamburger model construction 29 4.3. The localizing matrix construction of the Hamburger model 31 5.

Cauchy transforms arising from homomorphic conditional expectations parametrize noncommutative Pick functions

“…We note that Williams showed that the above theorem holds when E is a conditional expectation and ψ is an identity map if we assume additionally that f has some large analytic continuation at infinity corresponding to the classical compactly supported case [10]. Our result also generalizes previous results in [8,Section 5]. In the language of this paper, the representations established in the earlier setting held for B = C m .…”

Wandering Montel theorems for Hilbert space valued holomorphic functions

“…δ(x) < 1} is a non-commutative polynomial polyhedron. (We adopt the convention of [16] and write the tensors vertically for legibility.) Proving that the cones are closed is the key step in proving realization formulas for free holomorphic functions-see [2,1,7].…”

Aspects of non-commutative function theory

“…In addition to the motivations above, let us mention the work of Voiculescu [37], in the context of free probability; Popescu [26][27][28][29], in the context of extending classical function theory to d -tuples of bounded operators; Ball, Groenewald and Malakorn [8], in the context of extending realization formulas from functions of commuting operators to functions of non-commuting operators; Alpay and Kalyuzhnyi-Verbovetzkii [5] in the context of realization formulas for rational functions that are J -unitary on the boundary of the domain; and Helton, Klep and McCullough [13,14] and Helton and McCullough [18] in the context of developing a descriptive theory of the domains on which LMI and semi-definite programming apply; Muhly and Solel [23], in the context of tensorial function theory; Cimpric, Helton, McCullough and Nelson [10] in the context of non-commutative real Nullstellensätze; the second author and Timoney [22] and of Helton, Klep, McCullough and Slinglend, in [17] on non commutative automorphisms; and the work of Pascoe and TullyDoyle [25] on non-commutative operator monotonicity.…”