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In functional analysis, a reflexive operator algebra ''A'' is an operator algebra that has enough invariant subspaces to characterize it. Formally, ''A'' is reflexive if it is equal to the algebra of bounded operators which leave invariant each subspace left invariant by every operator in ''A''. This should not be confused with a reflexive space. == Examples == Nest algebras are examples of reflexive operator algebras. In finite dimensions, these are simply algebras of all matrices of a given size whose nonzero entries lie in an upper-triangular pattern. In fact if we fix any pattern of entries in an ''n'' by ''n'' matrix containing the diagonal, then the set of all ''n'' by ''n'' matrices whose nonzero entries lie in this pattern forms a reflexive algebra. An example of an algebra which is ''not'' reflexive is the set of 2 by 2 matrices : This algebra is smaller than the Nest algebra : but has the same invariant subspaces, so it is not reflexive. If ''T'' is a fixed ''n'' by ''n'' matrix then the set of all polynomials in ''T'' and the identity operator forms a unital operator algebra. A theorem of Deddens and Fillmore states that this algebra is reflexive if and only if the largest two blocks in the Jordan normal form of ''T'' differ in size by at most one. For example, the algebra : which is equal to the set of all polynomials in : and the identity is reflexive. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「reflexive operator algebra」の詳細全文を読む スポンサード リンク
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