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commutant lifting theorem : ウィキペディア英語版
commutant lifting theorem
In operator theory, the commutant lifting theorem, due to Sz.-Nagy and Foias, is a powerful theorem used to prove several interpolation results.
==Statement==
The commutant lifting theorem states that if ''T'' is a contraction on a Hilbert space ''H'', ''U'' is its minimal unitary dilation acting on some Hilbert space ''K'' (which can be shown to exist by Sz.-Nagy's dilation theorem), and ''R'' is an operator on ''H'' commuting with ''T'', then there is an operator ''S'' on ''K'' commuting with ''U'' such that
:R T^n = P_H S U^n \vert_H \; \forall n \geq 0,
and
:\Vert S \Vert = \Vert R \Vert.
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''.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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