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Matrix convex verbatim enumeration functions are graphical

J. E. Pascoe, Ryan Tully-Doyle

TL;DR

The paper develops a bridge between verbatim generating functions of Pythagorean languages and matrix convex analysis, showing that the verbatim enumeration function $f$ is matrix convex when the language is Pythagorean and admits a butterfly realization tied to a graph representation. A central link is the positive semidefinite Hankel-type matrix $C=[c_{\beta^*\alpha}]$, whose structure explains when $f$ corresponds to a generating function for a Pythagorean language via a Kraus/butterfly realization. It then connects these ideas to classical constructions (e.g., Ando unitarization, AMT) and to concrete combinatorial models (Dyck and Motzkin paths), culminating in a Gelfand-type numerical radius formula $w(Z)=\frac12\limsup_{n\to\infty}\|p_n(Z,Z^*)\|^{1/(2n)}$ where $Y(Z)=\sum_n p_n(Z,Z^*)$ enumerates irreducible Dyck paths. Together, the results fuse free noncommutative analysis, formal-language combinatorics, and operator theory, yielding new representations and computational tools for matrix-valued functions and boundary phenomena in several complex variables.

Abstract

We give a relation between verbatim generating functions of what we call Pythagorean languages and matrix convexity. Namely, several multivariate matrix convex functions occurring in the existing matrix analysis literature arise naturally in a combinatorial way. We give a Gelfand type formula for the numerical radius.

Matrix convex verbatim enumeration functions are graphical

TL;DR

The paper develops a bridge between verbatim generating functions of Pythagorean languages and matrix convex analysis, showing that the verbatim enumeration function is matrix convex when the language is Pythagorean and admits a butterfly realization tied to a graph representation. A central link is the positive semidefinite Hankel-type matrix , whose structure explains when corresponds to a generating function for a Pythagorean language via a Kraus/butterfly realization. It then connects these ideas to classical constructions (e.g., Ando unitarization, AMT) and to concrete combinatorial models (Dyck and Motzkin paths), culminating in a Gelfand-type numerical radius formula where enumerates irreducible Dyck paths. Together, the results fuse free noncommutative analysis, formal-language combinatorics, and operator theory, yielding new representations and computational tools for matrix-valued functions and boundary phenomena in several complex variables.

Abstract

We give a relation between verbatim generating functions of what we call Pythagorean languages and matrix convexity. Namely, several multivariate matrix convex functions occurring in the existing matrix analysis literature arise naturally in a combinatorial way. We give a Gelfand type formula for the numerical radius.
Paper Structure (4 sections, 6 theorems, 22 equations, 6 figures)

This paper contains 4 sections, 6 theorems, 22 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Matrix convex functions
  3. Some examples
  4. A Gelfand-type numerical radius formula

Key Result

Theorem 2.1

Let $f$ be matrix convex with a power series converging absolutely and uniformly on a neighborhood $D$ of $0$. Then there exist contractions $T_i$, vectors $Q_i$, a scalar $a_0$, and a continuous linear function $L$ so that

Figures (6)

  • Figure 1: Example of a Dyck path corresponding to the string "$[[[]][[]]]$".
  • Figure 2: Example of a Motzkin path, which corresponds to "$[xx[xx][]]$" in our string formulation.
  • Figure 3: The undirected loop graph on one vertex corresponds to the Schur complement of $\left[ ABCD -1 \right]$, that is, $A-B(D-1)^{-1}C$, where the edge is $D$-colored.
  • Figure 4: An infinite directed path graph corresponds to Ando's unitarization function used in analysis of the numerical radius. The Ando unitarization function satisfies $Y(Z) = Z(1-Y(Z))^{-1}Z^*.$ Note that the verbatim enumeration function gives the irreducible Dyck paths.
  • Figure 5: The two by two Anderson-Morley-Trapp function corresponds to an infinite path graph with self-loops added at each of the vertices. Note the verbatim enumeration function gives Motzkin paths. The Anderson-Morley-Trapp function satisfies $Y(Z,X) = Z(1-Y(Z,X)-X)^{-1}Z^*.$ In general, for larger sizes, we see the nodes are blown up into complete graphs.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Theorem 2.1: royal
  • Lemma 2.2: royal
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 4.1
  • Corollary 4.2