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On a minimal Andô dilation for a pair of strict contractions

Swapan Jana, Sourav Pal

Abstract

The isometric dilation of a pair of commuting contractions due to Andô is not minimal. We modify Andô's dilation and construct a minimal isometric dilation on $\mathcal H \oplus_2 \ell_2(\mathcal H \oplus_2 \mathcal H)$ for a commuting pair of strict contractions on a Hilbert space $\mathcal H$. In the same spirit, we construct under certain conditions a minimal Andô dilation for a commuting pair of strict Banach space contractions. Further, we show that an Andô dilation is possible even for a more general pair of commuting contractions $(T_1,T_2)$ on a normed space $\mathbb X$ provided that the function $A_{T_i}: \mathbb X \rightarrow \mathbb R$ given by $A_{T_i}(x)=(\|x\|^2-\|T_ix\|^2)^{\frac{1}{2}}$ defines a norm on $\mathbb X$ for $i=1,2$.

On a minimal Andô dilation for a pair of strict contractions

Abstract

The isometric dilation of a pair of commuting contractions due to Andô is not minimal. We modify Andô's dilation and construct a minimal isometric dilation on for a commuting pair of strict contractions on a Hilbert space . In the same spirit, we construct under certain conditions a minimal Andô dilation for a commuting pair of strict Banach space contractions. Further, we show that an Andô dilation is possible even for a more general pair of commuting contractions on a normed space provided that the function given by defines a norm on for .

Paper Structure

This paper contains 5 sections, 5 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Dilation for a pair of strict contractions
  3. Dilation in a more general setting
  4. Declarations
  5. Data availability statement

Key Result

Theorem 1.2

A strict contraction $T$ on a Banach space $\mathbb{X}$ dilates to a Banach space isometry if and only if the map $A_T:\mathbb{X} \to [0,\infty)$ defined by induces a norm on $\mathbb{X}$. Moreover, the corresponding minimal isometric dilation space of $T$ is isometrically isomorphic to $\mathbb{X}\oplus_2 \ell_2(\mathbb{X}_0)$, where $\mathbb{X}_0$ is the Banach space $(\mathbb{X}, A_T)$.

Theorems & Definitions (11)

  • Definition 1.1
  • Theorem 1.2: JPR, Theorem 7.13
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Theorem 3.1
  • ...and 1 more