Polynomial Controls on Unitary Laws and Their Energy

In the setting of The Real Numbers: Standing Notation and Background, let $\mathbb{N}$ be the natural numbers, $d\in\mathbb{N}$, $[d]$ the initial segment determined by $d$, and $\mathcal{L}_{d}$ the set of unitary laws of $d$-tuples. Word polynomials, their products and self-adjointness are those of Word Polynomials in Unitary Letters: the Normed *-Algebra of Finitely Supported Word Functions, and $\lambda(X)$ is the evaluation of a word polynomial $X$ by $\lambda\in\mathcal{L}_{d}$. Finite sums are those of Sum over a Finite Index Set.

1. (Polynomial controls)¶ A polynomial control is a $d$-tuple $a=(a^{1},\dots,a^{d})$ of self-adjoint word polynomials; $\mathcal{P}_{d}$ is the set of polynomial controls. It contains the tuple of zero word polynomials, the zero word polynomial being self-adjoint.

2. (Energy)¶ For $a\in\mathcal{P}_{d}$ and $\lambda\in\mathcal{L}_{d}$, the energy of $a$ under $\lambda$ is

$$\lVert a\rVert_{\lambda}^{2}=\sum_{i\in[d]}\lambda\bigl(a^{i}a^{i}\bigr),$$

a nonnegative real number: each term is one by Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound §positive, and a finite sum of nonnegative reals is nonnegative by Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §nonnegative. $\lVert a\rVert_{\lambda}$ is its nonnegative square root, given by Existence and Uniqueness of the Nonnegative Square Root.