Related papers: Free Pick functions: representations, asymptotic b…
We consider certain scalar product of symmetric functions which is parameterized by a function $r$ and an integer $n$. One the one hand we have a fermionic representation of this scalar product. On the other hand we get a representation of…
The main results presented in this paper provide a complete and explicit description of all solutions to the left tangential operator Nevanlinna- Pick interpolation problem assuming the associated Pick operator is strictly positive. The…
We survey recent results regarding the study of dynamical properties of the space of positive definite functions and characters of higher rank lattices. These results have several applications to ergodic theory, topological dynamics,…
Motivated by the asymptotic collective behavior of random and deterministic matrices, we propose an approximation (called "free deterministic equivalent") to quite general random matrix models, by replacing the matrices with operators…
The motivation behind this paper is threefold. Firstly, to study, characterize and realize operator concavity along with its applications to operator monotonicity of free functions on operator domains that are not assumed to be matrix…
We show how the combination of new "linearization" ideas in free probability theory with the powerful "realization" machinery -- developed over the last 50 years in fields including systems engineering and automata theory -- allows solving…
We generalize our earlier results from \cite{K} on the Bessmertny\u{\i} class of operator-valued functions holomorphic in the open right poly-halfplane which admit representation as a Schur complement of a block of a linear homogeneous…
We study the questions of determining the asymptotics of the probabilistic characteristics of additive arithmetic functions in the paper, regardless of whether they have a limit distribution or not. Several assertions are proved about the…
We analyze the representation of $A^{n}$ as a linear combination of $A^{j},\ 0\leq j\leq k-1,$ where $A$ is a $k\times k$ matrix. We obtain a first order asymptotic approximation of $A^{n}$ as $n\to\infty,$ without imposing any special…
In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the…
The class of operator-valued functions which are homogeneous of degree one, holomorphic in the open right polyhalfplane, have positive semidefinite real parts there and take selfadjoint operator values at real points, and its subclass…
For a Hilbert function space $\mathcal H$ the Smirnov class $\mathcal N^+(\mathcal H)$ is defined to be the set of functions expressible as a ratio of bounded multipliers of $\mathcal H$, whose denominator is cyclic for the action of…
In this paper, we introduce new classes of functions that extend the known classes of functions of complex variable, such as entire functions, meromorphic functions, rational functions and polynomial functions and take values in the set of…
We generalize module weak-* Haagerup tensor products to obtain complete quotients of normal Haagerup tensor product included in canonical Hilbert spaces associated to completely positive normal (covariance) maps $\eta$ on a finite von…
We study the category $\mathcal{A}$ of smooth semilinear representations of the infinite symmetric group over the field of rational functions in infinitely many variables. We establish a number of results about the structure of…
We carry out the asymptotic analysis as $n \to \infty$ of a class of orthogonal polynomials $p_{n}(z)$ of degree $n$, defined with respect to the planar measure \begin{equation*} d\mu(z) = (1-|z|^{2})^{\alpha-1}|z-x|^{\gamma}\mathbf{1}_{|z|…
We show "free theorems" in the style of Wadler for polymorphic functions in homotopy type theory as consequences of the abstraction theorem. As an application, it follows that every space defined as a higher inductive type has the same…
Andrews-Dyson-Hickerson, Cohen build a striking relation between q-hypergeometric series, real quadratic fields, and Maass forms. Thanks to the works of Lewis-Zagier and Zwegers we have a complete understanding on the part of these…
We study the distribution of partition parts in arithmetic progressions and find asymptotic results that capture all exponentially growing terms. This is accomplished by studying the behavior of non-modular Eisenstein series that appear in…
We characterize asymptotic collective behaviour of rectangular random matrices, the sizes of which tend to infinity at different rates: when embedded in a space of larger square matrices, independent rectangular random matrices are…