Operator theory Invariant subspaces and dilations Theorem

Commutant lifting theorem

Builds on Hilbert space Orthogonal projection Isometry Hardy space Inner function

Statement

Let $H$ be a Hilbert space 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. 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 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 $H^2$ to a model space $H^2 \ominus \theta H^2$ , with $\theta$ an inner function. 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.

Going deeper

For a student

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, which provides the minimal isometric dilation.

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Tag	`3TVXB7X266`
URI	`m2://3TVXB7X266`
Kind	statement, theorem
Writing	W0 stub
MSC	47A20, 47A57
Status	This copy of the site has no build data. Run `m2 status 3TVXB7X266`.
About	Isometry
Uses	Hilbert space, Orthogonal projection, Isometry, Hardy space, Inner function
Same as	Wikidata Q5155083 Wikipedia Commutant lifting theorem
Changed	2 November 2026

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