---
tag: 7KS1W5K54S
kind: statement
role: theorem
name: von Neumann's inequality
msc: [47A63, 47A20]
uses: [HM2HHE3J51, 1G0B73WPX8]
about: [HM2HHE3J51]
sources:
  - url: https://en.wikipedia.org/wiki/Von_Neumann%27s_inequality
same_as:
  wikidata: Q7941493
  wikipedia: Von Neumann's inequality
---

## Statement

Let $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $T \colon H \to H$ be a bounded linear operator with $\norm{T} \le 1$.
Then for every [[polynomial|1G0B73WPX8]] $p$ with complex coefficients,
$$\norm{p(T)} \le \sup \set{\abs{p(z)} : z \in \C, \ \abs{z} \le 1}.$$

## Remarks

Proof idea: by [[Sz.-Nagy's dilation theorem|YB7Q3EK0ND]] there is a unitary $U$ on a larger space $K \supseteq H$ with $T^n = P_H U^n|_H$ for $n \ge 0$, so $p(T) = P_H \, p(U)|_H$.
Hence $\norm{p(T)} \le \norm{p(U)}$.
Since $U$ is unitary, its spectrum lies on the unit circle.
So $\norm{p(U)}$ is the maximum of $\abs{p}$ on the spectrum, which is at most the maximum of $\abs{p}$ on the closed unit disc.

Example: for the nilpotent contraction $T = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $p(z) = 1 + z$, one has $p(T) = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ with norm $\tfrac{1 + \sqrt 5}{2} \approx 1.618$, which is at most $\max_{\abs{z} \le 1} \abs{1 + z} = 2$.

The Hilbert space structure matters.
On a Banach space a contraction need not satisfy the inequality.
The inequality extends to two commuting contractions (Andô's theorem) but fails in general for three commuting contractions.

## Explanations

### L1 human

If an operator never stretches vectors, then a polynomial in the operator is no bigger than the same polynomial is on the unit disc of the complex plane.
So a contraction behaves, for polynomial expressions, no worse than a complex number of modulus at most $1$.

### L2 human

The inequality lets one define $f(T)$ for every $f$ in the disc algebra, the uniform closure of the polynomials on the closed disc, with $\norm{f(T)} \le \norm{f}_\infty$.
This is the start of the holomorphic functional calculus of contractions developed by Sz.-Nagy and Foias.
Its failure for three commuting contractions shows that multivariable operator theory differs from the one-variable theory.