Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation and Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, let $n\in\mathbb{N}$, let $H$ be a complex Hilbert space and let $T=(T_{1},\dots,T_{n})$ be an $n$-tuple in $\mathcal{L}(H)$. Composites of elements of $\mathcal{L}(H)$ lie in $\mathcal{L}(H)$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations, so composition $(A,B)\mapsto AB$ is a binary operation on $\mathcal{L}(H)$.

1. (Products along a word)¶ For $w\in W_{n}$ the operator $T_{w}\in\mathcal{L}(H)$ is defined by $T_{\varnothing}=I$ and, if $w$ has length $k\in\mathbb{N}$,

meaning the value at $k$ of the unique map $\pi:[k]\to\mathcal{L}(H)$ with $\pi(1)=T_{w_{1}}$ and $\pi(i+1)=\pi(i)\,T_{w_{i+1}}$ whenever $i+1\in[k]$, given by Existence and Uniqueness of Iterates of a Binary Operation for composition.

2. (Evaluation)¶ The evaluation at $T$ is the unique linear map $\mathcal{P}_{n}\to\mathcal{L}(H)$, $p\mapsto p(T)$, with $x_{w}(T)=T_{w}$ for every $w\in W_{n}$, which exists and is unique by Linear Maps from Noncommutative Polynomials into Bounded Operators: Existence and Uniqueness from Values on Monomials §extension. The operator $p(T)$ is the value of $p$ at $T$.