---
tag: 3TVXB7X266
kind: statement
role: theorem
name: Commutant lifting theorem
msc: [47A20, 47A57]
uses: [HM2HHE3J51, 0B8KZRT28M, 02HPP5VCZH, PNNNFJVF6W, NN8XH5RJYB]
about: [02HPP5VCZH]
sources:
  - url: https://en.wikipedia.org/wiki/Commutant_lifting_theorem
same_as:
  wikidata: Q5155083
  wikipedia: Commutant lifting theorem
---

## Statement

Let $H$ be a [[Hilbert space|HM2HHE3J51]] and let $T \colon H \to H$ be a contraction, that is, a bounded linear operator with $\norm{T} \le 1$.
Let $V \colon K \to K$ be a minimal isometric dilation of $T$.
This means that $K$ is a Hilbert space containing $H$ as a closed subspace and $V$ is an [[isometry|02HPP5VCZH]].
It also means that $T^n = P_H V^n|_H$ for all $n \ge 0$, and that $K$ is the closed span of $\bigcup_{n \ge 0} V^n H$.
Here $P_H$ is the [[orthogonal projection|0B8KZRT28M]] of $K$ onto $H$.
Let $A \colon H \to H$ be a bounded linear operator with $AT = TA$.
Then there is a bounded linear operator $B \colon K \to K$ such that
$$BV = VB, \qquad \norm{B} = \norm{A}, \qquad P_H B = A P_H.$$

## Remarks

The condition $P_H B = A P_H$ says that $B^*$ maps $H$ into $H$ and $B^*|_H = A^*$; so $B$ lifts $A$ to the dilation space without increasing the norm.
The proof constructs $B$ step by step on the pieces of $K$, with a one-step extension lemma for operator matrices that keeps the norm.
The theorem is due to Sz.-Nagy and Foias, and Sarason proved an earlier special case.

Example: take $T$ to be the compression of the shift $S$ on the [[Hardy space|PNNNFJVF6W]] $H^2$ to a model space $H^2 \ominus \theta H^2$, with $\theta$ an [[inner function|NN8XH5RJYB]].
Then the lifts of operators commuting with $T$ commute with $S$, so they are multiplications by bounded analytic functions.
This gives Sarason's theorem: each operator commuting with $T$ is the compression of multiplication by some $\varphi \in H^\infty$ with the same norm.

## Explanations

### L1 human

A contraction can be seen as a piece of an isometry acting on a larger space.
The theorem says that any operator that commutes with the contraction can be extended to the larger space so that it commutes with the isometry, with no loss in norm.

### L2 human

Commutant lifting gives a unified operator-theoretic approach to interpolation problems.
The Nevanlinna–Pick and Carathéodory–Fejér problems ask for a bounded analytic function on the disc with prescribed values or Taylor coefficients and least supremum norm.
Both reduce to lifting an operator that commutes with a compressed shift.
The theorem builds on [[Sz.-Nagy's dilation theorem|YB7Q3EK0ND]], which provides the minimal isometric dilation.