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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)
##### Keywords:
linear relation; conjugate; linear operator
Zbl 0348.47021
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