commutant lifting theorem

proper noun

A theorem in operator theory, stating that, if T is a contraction on a Hilbert space H, and U is its minimal unitary dilation acting on some Hilbert space K, and R is an operator on H commuting with T, then there is an operator S on K commuting with U such that RTⁿ=P_HSUⁿ|_H;∀n≥0, and ‖S‖=‖R‖. In other words, an operator from the commutant of T can be "lifted" to an operator in the commutant of the unitary dilation of T.

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Meanings

proper nounthe commutant lifting theorem
  1. mathematicsA theorem in operator theory, stating that, if T is a contraction on a Hilbert space H, and U is its minimal unitary dilation acting on some Hilbert space K, and R is an operator on H commuting with T, then there is an operator S on K commuting with U such that RTⁿ=P_HSUⁿ|_H;∀n≥0, and ‖S‖=‖R‖. In other words, an operator from the commutant of T can be "lifted" to an operator in the commutant of the unitary dilation of T.
Cite this entry
APA

commutant lifting theorem. (n.d.). In WatkinsX Dictionary. Retrieved November 5, 2026, from https://dictionary.watkinsx.com/word/commutant%20lifting%20theorem

MLA

“commutant lifting theorem.” WatkinsX Dictionary, https://dictionary.watkinsx.com/word/commutant%20lifting%20theorem. Accessed November 5, 2026.

Chicago

WatkinsX Dictionary, s.v. “commutant lifting theorem,” accessed November 5, 2026, https://dictionary.watkinsx.com/word/commutant%20lifting%20theorem.

Source: commutant lifting theorem on Wiktionary, CC BY-SA 4.0.