An inductive Julia-Caratheodory theorem for Pick functions in two variables

We study the asymptotic behavior of Pick functions, analytic functions which take the upper half plane to itself. We show that if a two variable Pick function $f$ has real residues to order $2N-1$ at infinity and the imaginary part of the remainder between $f$ and this expansion is of order $2N+1,$ then $f$ has real residues to order $2N$ and directional residues to order $2N+1.$ Furthermore, $f$ has real residues to order $2N+1$ if and only if the $2N+1$-th derivative is given by a polynomial, thus obtaining a two variable analogue of a higher order Julia-Carath\'eodory type theorem

The inverse function theorem and the resolution of the Jacobian conjecture in free analysis

We establish an invertibility criterion for free polynomials and free functions evaluated on some tuples of matrices. We show that if the derivative is nonsingular on some domain closed with respect to direct sums and similarity, the function must be invertible. Thus, as a corollary, we establish the Jacobian conjecture in this context. Furthermore, our result holds for commutative polynomials evaluated on tuples of commuting matrices

Positivstellensatz\"e for noncommutative rational expressions

We derive some Positivstellensatz\"e for noncommutative rational expressions from the Positivstellensatz\"e for noncommutative polynomials. Specifically, we show that if a noncommutative rational expression is positive on a polynomially convex set, then there is an algebraic certificate witnessing that fact. As in the case of noncommutative polynomials, our results are nicer when we additionally assume positivity on a convex set-- that is, we obtain a so-called "perfect Positivstellensatz" on convex sets.Comment: 6 page

The wedge-of-the-edge theorem: edge-of-the-wedge type phenomenon within the common real boundary

The edge-of-the-wedge theorem in several complex variables gives the analytic continuation of functions defined on the poly upper half plane and the poly lower half plane, the set of points in $\mathbb{C}^d$ with all coordinates in the upper and lower half planes respectively, through a set in real space, $\mathbb{R}^d.$ The geometry of the set in the real space can force the function to analytically continue within the boundary itself, which is qualified in our wedge-of-the-edge theorem. For example, if a function extends to the union of two cubes in $\mathbb{R}^d$ which are positively oriented, with some small overlap, the functions must analytically continue to a neighborhood of that overlap of a fixed size not depending of the size of the overlap.Comment: 12 page

On the continuation of locally operator monotone functions

We generalize the phenomenon of continuation from complex anal- ysis to locally operator monotone functions. Along the lines of the egde-of- the-wedge theorem, we prove continuations exist dependent only on geometric features of the domain and, namely, independent of the function values. We prove a generalization of the Julia inequality for a class of functions containing locally operator monotone functions, Pick functions

The noncommutative L\"owner theorem for matrix monotone functions over operator systems

Given a function $f: (a,b) \rightarrow \mathbb{R},$ L\"owner's theorem states $f$ is monotone when extended to self-adjoint matrices via the functional calculus, if and only if $f$ extends to a self-map of the complex upper half plane. In recent years, several generalizations of L\"owner's theorem have been proven in several variables. We use the relaxed Agler, McCarthy and Young theorem on locally matrix monotone functions in several commuting variables to generalize results in the noncommutative case. Specifically, we show that a real free function defined over an operator system must analytically continue to a noncommutative upper half plane as map into another noncommutative upper half plane.Comment: 7 page

Committee spaces and the random column-row property

A committee space is a Hilbert space of power series, perhaps in several or noncommuting variables, such that $\|z^\alpha\|\|z^\beta\| \geq \|z^{\alpha+\beta}\|.$ Such a space satisfies the true column-row property when ever the map transposing a column multiplier to a row multiplier is contractive. We describe a model for random multipliers and show that such random multipliers satisfy the true column-row property. We also show that the column-row property holds asymptotically for large random Nevanlinna-Pick interpolation problems where the nodes are chosen identically and independently. These results suggest that finding a violation of the true column-row property for the Drury-Arveson space via na\"ive random search is unlikely.Comment: 10 page

Cauchy transforms arising from homomorphic conditional expectations parametrize free Pick functions but those arising from conditional expectations do not

Nevanlinna showed that Cauchy transforms of probability measures parametrize all functions from the upper half plane into itself satisfying a certain asymptotic condition at infinity. We show that the correspondence fails in general for the unbounded case for somewhat trivial reasons; however, we show that in a setting of "homomorphic" operator valued free probability that Cauchy transforms of homomorphic conditional expectations parametrize free Pick functions.Comment: 12 page

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

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. Loewner connected the Pick class to matrix monotone functions. We generalize the Nevanlinna representation theorems and Loewner'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.Comment: 53 page

Escaping nontangentiality: Towards a controlled tangential amortized Julia-Carath\'eodory theory

Let $f: D \rightarrow \Omega$ be a complex analytic function. The Julia quotient is given by the ratio between the distance of $f(z)$ to the boundary of $\Omega$ and the distance of $z$ to the boundary of $D.$ A classical Julia-Carath\'eodory type theorem states that if there is a sequence tending to $\tau$ in the boundary of $D$ along which the Julia quotient is bounded, then the function $f$ can be extended to $\tau$ such that $f$ is nontangentially continuous and differentiable at $\tau$ and $f(\tau)$ is in the boundary of $\Omega.$ We develop an extended theory when $D$ and $\Omega$ are taken to be the upper half plane which corresponds to amortized boundedness of the Julia quotient on sets of controlled tangential approach, so-called $\lambda$-Stolz regions, and higher order regularity, including but not limited to higher order differentiability, which we measure using $\gamma$-regularity. Applications are given, including perturbation theory and moment problems.Comment: 26 page