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On factorization of the shift semigroup

Tirthankar Bhattacharyya, Shubham Rastogi, Kalyan B. Sinha, Vijaya Kumar U

TL;DR

This work completely characterizes all factorizations of the right shift semigroup $\\mathscr S^\\mathcal{E}$ on $L^2(\\mathbb{R}_+,\\mathcal{E})$ into a product of $n$ commuting contractive semigroups when $\\mathcal{E}$ is finite-dimensional. The authors leverage the Berger–Coburn–Lebow model for commuting isometries, Cooper’s semigroup decomposition, and a commutant analysis on a vector-valued Hardy space to reduce factorization to structured inner-operator data, framed by the Herglotz extreme-point analysis. They prove that factorizations correspond to pairs $(\\underline{A},\\underline{B})$ of self-adjoint and positive-contraction data on $\\mathcal{E}$, with explicit sum and commutativity constraints, yielding semigroups of the form $V_{j,t}=M_{e^{t(iA_j+\\zeta(z)B_j)}}$. In the multiplicity-one case, the factors are completely classified as $V_{j,t}=e^{i a_j t}S_{b_j t}$ with $\\sum a_j=0$ and $\\sum b_j=1$, parametrized by $(\\underline{a},\\underline{b})$ and associated Blaschke-type inner functions. Overall, the paper provides a finite-dimensional parameterization of all factorizations linking operator model theory to inner-function data via a convex-extremal approach in the Herglotz class.

Abstract

Let $\E$ be a finite dimensional Hilbert space. This note finds all factorizations of the right shift semigroup $§^\E=(S_t^\E)_{t\ge 0}$ on $L^2(\R_+,\E)$ into the product of $n$ commuting contractive semigroups, i.e., characterizes all $n$-tuples of commuting semigroups $(\V_1,\V_2,...,\V_n)$ where $\V_i=(V_{i,t})_{t\ge 0}$ for $i=1,2,...,n$ are semigroups of contractions satisfying $V_{i,t}V_{j,t}=V_{j,t}V_{i,t}$ for all $i$ and $j$ and $S_t^\E=V_{1,t}V_{2,t}\cdots V_{n,t}$ for all $t\ge 0.$ The factorizations are characterized by tuples of self-adjoint operators $\underline{A}=(A_1,A_2,...,A_n)$ and tuples of positive contractions $\underline{B}=(B_1,B_2,...,B_n)$ on $\E$ satisfying certain conditions which are stated in \cref{thm:psi12}. One of the tools of our analysis is a convexity argument using the extreme points of the {\em Herglotz } class of functions \[P:=\{f:\D\to \C \text{ is analytic}, \Re{f}>0 \text{ and }f(0)=1 \}.\]

On factorization of the shift semigroup

TL;DR

This work completely characterizes all factorizations of the right shift semigroup on into a product of commuting contractive semigroups when is finite-dimensional. The authors leverage the Berger–Coburn–Lebow model for commuting isometries, Cooper’s semigroup decomposition, and a commutant analysis on a vector-valued Hardy space to reduce factorization to structured inner-operator data, framed by the Herglotz extreme-point analysis. They prove that factorizations correspond to pairs of self-adjoint and positive-contraction data on , with explicit sum and commutativity constraints, yielding semigroups of the form . In the multiplicity-one case, the factors are completely classified as with and , parametrized by and associated Blaschke-type inner functions. Overall, the paper provides a finite-dimensional parameterization of all factorizations linking operator model theory to inner-function data via a convex-extremal approach in the Herglotz class.

Abstract

Let be a finite dimensional Hilbert space. This note finds all factorizations of the right shift semigroup on into the product of commuting contractive semigroups, i.e., characterizes all -tuples of commuting semigroups where for are semigroups of contractions satisfying for all and and for all The factorizations are characterized by tuples of self-adjoint operators and tuples of positive contractions on satisfying certain conditions which are stated in \cref{thm:psi12}. One of the tools of our analysis is a convexity argument using the extreme points of the {\em Herglotz } class of functions
Paper Structure (5 sections, 22 theorems, 77 equations, 1 figure)

This paper contains 5 sections, 22 theorems, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The B-C-L theorem
  3. The right shift semigroup and its commutant
  4. Factorization of the right shift semigroup
  5. The case of multiplicity one

Key Result

Theorem 1.1

[thm]Cooper Let $\mathscr V=(V_t)_{t\ge 0}$ be a one-parameter $C_0$-semigroup of isometries on a Hilbert space $\mathcal{H}.$ Then $\mathcal{H}$ uniquely decomposes as the direct sum of reducing subspaces $\mathcal{H}_p$ and $\mathcal{H}_u$ of $\mathscr V,$ so that Moreover, any c.n.u one-parameter isometric semigroup is unitarily equivalent to the shift semigroup $\mathscr S^\mathcal{E}=(S_t^\

Figures (1)

  • Figure 1: A Stolz type region $\{ z \in \mathbb D : |1 - z|^2 < M(1 - |z|^2) \}.$

Theorems & Definitions (50)

  • Theorem 1.1: Cooper Cooper
  • Definition 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • ...and 40 more