The Free Unitary Heat Generator: Bound, Continuity and Strict Dissipation in the Length-Weighted Gauge

In the setting of The Real Numbers: Standing Notation and Background, let $\mathbb{N}$ be the natural numbers and $d\in\mathbb{N}$. Let $W_{2d}$, the lengths $|w|$ and the set $W^{\circ}_{d}$ of cyclically reduced words be as in Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words and Reduced and Cyclically Reduced Words in Unitary Letters, let $\mathcal{L}_{d}$ be the set of unitary laws of $d$-tuples, let $\Theta$ be the free unitary heat generator, and let the gauge space $E_{d}$ with its norm $\lVert\cdot\rVert_{d}$ and inner product $\langle\cdot,\cdot\rangle_{d}$ and the length-weighted gauge $\lVert\cdot\rVert_{d,1}$ be as in The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge, and the embedding $\iota_{d}$ and the gauge distance $d_{\mathcal{L}}$ as in The Embedding of Unitary Laws into the Word Gauge Space and the Gauge Distance; by The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §hilbert $E_{d}$ is a real Hilbert space. Let $\mathbb{C}$ be the complex numbers with modulus $|z|$. For $\lambda\in\mathcal{L}_{d}$ let $\widehat{\Theta}\lambda:W^{\circ}_{d}\to\mathbb{C}$ be the restriction of $\Theta\lambda$ to $W^{\circ}_{d}$.

Then the following hold.

1. (Bound)¶ For every $\lambda\in\mathcal{L}_{d}$ and every $w\in W_{2d}$, $|\Theta\lambda(w)|\le|w|^{2}$. Consequently $\widehat{\Theta}\lambda\in E_{d}$ for every $\lambda\in\mathcal{L}_{d}$.

2. (Continuity)¶ The map $\lambda\mapsto\widehat{\Theta}\lambda$ is continuous from the metric space $(\mathcal{L}_{d},d_{\mathcal{L}})$ to $E_{d}$ with the metric $(x,y)\mapsto\lVert x-y\rVert_{d}$.

3. (Dissipation)¶ For all $\mu,\nu\in\mathcal{L}_{d}$,

$$\bigl\langle\iota_{d}(\mu)-\iota_{d}(\nu),\,\widehat{\Theta}\mu-\widehat{\Theta}\nu\bigr\rangle_{d}\le-\frac14\,\lVert\iota_{d}(\mu)-\iota_{d}(\nu)\rVert_{d,1}^{2},$$

where $\lVert\iota_{d}(\mu)-\iota_{d}(\nu)\rVert_{d,1}$ is defined by The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §sobolev.