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The conjugate of a product of linear relations. (English) Zbl 1150.47002
Let \(X\) and \(Y\) be normed linear spaces and \(T\) be a linear relation from \(X\) into \(Y\), i.e., a set-valued map \(T\: X\to 2^Y\) (\(Tx=\emptyset \) being also allowed). Its conjugate \(T'\) from \(Y'\) into \(X'\) is defined by \(x'\in T'y'\iff y'y-x'x=0\) \(\forall x\in X\) \(\forall y\in Tx\). The author gives necessary and sufficient conditions ensuring that \((ST)'= T'S'\). This extends the analogous results of K.-H. Förster and E.-O. Liebetrau [Stud. Math. 59, 301–306 (1977; Zbl 0348.47021)] for operators. Various useful properties of linear relations are established along the way.

MSC:
47A06 Linear relations (multivalued linear operators)
Citations:
Zbl 0348.47021
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