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linear relations; conjugates; linear operators
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.
[CG70] van Casteren J.A.W., Goldberg S.: The conjugate of a product of operators. Studia Math. 38 (1970), 125-130. MR 0275192
[Cro98] Cross R.W.: Multivalued Linear Operators. Marcel Dekker, New York, 1998. MR 1631548 | Zbl 0911.47002
[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
[For74] Förster K.-H.: Relativ co-stetige Operatoren in normierten Räumen. Arch. Math. 25 (1974), 639-645. MR 0397459
[Kaa64] Kaashoek M.A.: Closed linear operators on Banach spaces. Ph.D. Thesis, Univ. Leiden, 1964. MR 0185451 | Zbl 0138.07502
[Kas68] Kascic M.J.: Polynomials in linear relations. Pacific J. Math. 24 (1968), 291-295. MR 0222670 | Zbl 0155.19004
[Kat66] Kato T.: Perturbation Theory for Linear Operators. Grundlehren, vol. 132, Springer, Berlin, 1966. Zbl 0836.47009
[SN51] Sz.-Nagy B.: Perturbations des transformations linéaires fermées. Acta Sci. Math. Szeged 14 (1951), 125-137. MR 0047254 | Zbl 0045.21601
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