|
In mathematics, a Hilbert–Schmidt operator, named for David Hilbert and Erhard Schmidt, is a bounded operator ''A'' on a Hilbert space ''H'' with finite Hilbert–Schmidt norm : for and the Schatten norm of for ''p=2''. In Euclidean space is also called Frobenius norm, named for Ferdinand Georg Frobenius. The product of two Hilbert–Schmidt operators has finite trace class norm; therefore, if ''A'' and ''B'' are two Hilbert–Schmidt operators, the Hilbert–Schmidt inner product can be defined as : The Hilbert–Schmidt operators form a two-sided *-ideal in the Banach algebra of bounded operators on ''H''. They also form a Hilbert space, which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces : where ''H *'' is the dual space of ''H''. The set of Hilbert–Schmidt operators is closed in the norm topology if, and only if, ''H'' is finite-dimensional. An important class of examples is provided by Hilbert–Schmidt integral operators. Hilbert–Schmidt operators are nuclear operators of order 2, and are therefore compact. ==References== 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「Hilbert–Schmidt operator」の詳細全文を読む スポンサード リンク
|