# Article

 Title: The conjugate of a product of linear relations  (English) Author: Jaftha, J. J. Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 47 Issue: 2 Year: 2006 Pages: 265-273 . Category: math . Summary: Let $X$, $Y$ and $Z$ be normed linear spaces with $T(X\rightarrow Y)$ and $S(Y\rightarrow Z)$ linear relations, i.e. setvalued maps. We seek necessary and sufficient conditions that would ensure that $(ST)'=T'S'$. First, we cast the concepts of relative boundedness and co-continuity in the set valued case and establish a duality. This duality turns out to be similar to the one that exists for densely defined linear operators and is then used to establish the necessary and sufficient conditions. These conditions are similar to those for the single valued case. In the process, the well known characterisation of relativeboundedness for closed linear operators by Sz.-Nagy is extended to the multivalued linear maps and we compare our results to other known necessary and sufficient conditions. Keyword: linear relations Keyword: conjugates Keyword: linear operators MSC: 47A05 MSC: 47A06 idZBL: Zbl 1150.47002 idMR: MR2241531 . Date available: 2009-05-05T16:57:16Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119591 . Reference: [CG70] van Casteren J.A.W., Goldberg S.: The conjugate of a product of operators.Studia Math. 38 (1970), 125-130. MR 0275192 Reference: [Cro98] Cross R.W.: Multivalued Linear Operators.Marcel Dekker, New York, 1998. Zbl 0911.47002, MR 1631548 Reference: [FL77] Förster K.-H., Liebetrau E.-O.: On semi-Fredholm operators and the conjugate of a product of operators.Studia Math. 59 (1976/77), 301-306. MR 0435883 Reference: [For74] Förster K.-H.: Relativ co-stetige Operatoren in normierten Räumen.Arch. Math. 25 (1974), 639-645. MR 0397459 Reference: [Kaa64] Kaashoek M.A.: Closed linear operators on Banach spaces.Ph.D. Thesis, Univ. Leiden, 1964. Zbl 0138.07502, MR 0185451 Reference: [Kas68] Kascic M.J.: Polynomials in linear relations.Pacific J. Math. 24 (1968), 291-295. Zbl 0155.19004, MR 0222670 Reference: [Kat66] Kato T.: Perturbation Theory for Linear Operators.Grundlehren, vol. 132, Springer, Berlin, 1966. Zbl 0836.47009 Reference: [SN51] Sz.-Nagy B.: Perturbations des transformations linéaires fermées.Acta Sci. Math. Szeged 14 (1951), 125-137. Zbl 0045.21601, MR 0047254 .

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