Related papers: The Free Functional Calculus in General
Working over a field $k$ of characteristic zero, the category of analytic contravariant functors on the category of finitely-generated free groups is shown to be equivalent to the category of representations of the $k$-linear category…
The goal of the present paper is to introduce and study noncommutative Hardy spaces associated with the regular $\Lambda$-polyball, to develop a functional calculus on noncommutative Hardy spaces for the completely non-coisometric (c.n.c.)…
A natural connection between rational functions of several real or complex variables, and subspace collections is explored. A new class of function, superfunctions, are introduced which are the counterpart to functions at the level of…
We develop a theory of holomorphic functions in several noncommuting (free) variables and thus provide a framework for the study of arbitrary n-tuples of operators. The main topics are the following: Free holomorphic functions and Hausdorff…
We discuss free probability theory and free harmonic analysis from a categorical perspective. In order to do so, we extend first the set of analytic convolutions and operations and then show that the comonadic structure governing free…
We study the generalization of shifted Jack polynomials to arbitrary multiplicity free spaces. In a previous paper (math.RT/0006004) we showed that these polynomials are eigenfunctions for commuting difference operators. Our central result…
We want to replace categories, functors and natural transformations by categories, open functors and open natural transformations. In analogy with open dynamical systems, the adjective open is added here to mean that some external…
The functional equation defining the free cumulants in free probability is lifted successively to the noncommutative Fa\`a di Bruno algebra, and then to the group of a free operad over Schr\"oder trees. This leads to new combinatorial…
We establish a theory of NC functions on a class of von Neumann algebras with a particular direct sum property, e.g. $B(\mathcal{H})$. In contrast to the theory's origins, we do not rely on appealing to results from the matricial case. We…
We study polynomial functors over locally cartesian closed categories. After setting up the basic theory, we show how polynomial functors assemble into a double category, in fact a framed bicategory. We show that the free monad on a…
We formulate and prove a free quantum analogue of the first fundamental theorems of invariant theory. More precisely, the polynomial functions algebras are replaced by free algebras, while the universal cosovereign Hopf algebras play the…
We introduce a class of Schur type functions associated with polynomial sequences of binomial type. This can be regarded as a generalization of the ordinary Schur functions and the factorial Schur functions. This generalization satisfies…
We define an enumerative function F(n,k,P,m) which is a generalization of binomial coefficients. Special cases of this function are also power function, factorials, rising factorials and falling factorials. The first section of the paper is…
A rational function is the ratio of two complex polynomials in one variable without common roots. Its degree is the maximum of the degrees of the numerator and the denominator. Rational functions belong to the same class if one turns into…
A combinatorial formula is derived which expresses free cumulants in terms of classical comulants. As a corollary, we give a combinatorial interpretation of free cumulants of classical distributions, notably Gaussian and Poisson…
We construct a generalization of the theory of symmetric functions involving functions of commuting and anticommuting (Grassmannian) variables. These new functions, called symmetric functions in superspace, are invariant under the diagonal…
Polynomial functors are a categorical generalization of the usual notion of polynomial, which has found many applications in higher categories and type theory: those are generated by polynomials consisting a set of monomials built from sets…
Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on…
Starting from a differential equation for the unique field, the equation for the generating vector |V> of the n-point information (correlation and smeared functions) in the free Fock space is derived. In derived equation, due to appropriate…
Let $A$ be a rational function. For any decomposition of $A$ into a composition of rational functions $A=U\circ V$ the rational function $\widetilde A=V\circ U$ is called an elementary transformation of $A$, and rational functions $A$ and…