English Dictionary

commutant lifting theorem

固有名詞

(mathematics) 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.

日本語の意味

(数学/演算子論) ヒルベルト空間上の縮小 T に対して、T と可換な演算子 R が存在すると、その可換性を保ったまま、T の最小ユニタリ拡大 U に作用する対応する演算子 S（‖S‖ = ‖R‖）が存在することを示す定理。

In my research I applied the commutant lifting theorem to extend an operator commuting with a contraction to its minimal unitary dilation.

私の研究では、縮小作用素と可換する作用素を最小ユニタリ拡大に持ち上げる定理を応用して、そのような作用素を拡張しました。

意味（1）

(mathematics) 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.