Theorem 41.1.8 (Von Neumann’s Inequality for Matrices).label Let $n \in \natp$, $T \in M_{n}(\complex)$ with $\norm{T}_{M_n(\complex)}\le 1$, and $U, V, S \in M_{n}(\complex)$ such that:

1. (a)

   $U, V$ are unitary.

2. (b)

   $S = \text{diag}(s_{1}, \cdots, s_{n})$, where $\seqf{s_j}\subset [0, 1]$.

Let $B = \ol{B_\complex(0, 1)}^{n} \subset \complex^{n}$ and

and $p \in \complex[z]$, then:

1. (1)

   For each $\xi, \eta \in \complex^{n}$, let

   \[f_{\xi, \eta}: B \to \complex \quad (z_{1}, \cdots, z_{n}) \mapsto \dpn{p(T(z_1, \cdots, z_n))\xi, \eta}{\complex^n}\]

   
   

   then $f_{\xi, \eta}$ achieves its maximum modulus on $\partial B$.

2. (2)

   For any $(z_{1}, \cdots, z_{n}) \in B$, if

   \[\normn{p(T(z_1, \cdots, z_n))}_{M_n(\complex)}= \max_{w \in \overline{B}}\normn{p(T(w_1, \cdots, w_n))}_{M_n(\complex)}\]

   
   

   then $T(z_{1}, \cdots, z_{n})$ is unitary.

3. (3)

   $\norm{p(T)}_{M_n(\complex)}\le \max_{W \in U(M_n(\complex))}\norm{p(W)}_{M_n(\complex)}$.

4. (4)

   $\norm{p(T)}_{M_n(\complex)}\le \norm{p}_{H^\infty(D)}$.

For each $\seqf{z}\in \overline{B_{\complex^n}(0, 1)}$