# The conjugate of a product of linear relations

Commentationes Mathematicae Universitatis Carolinae (2006)

- Volume: 47, Issue: 2, page 265-273
- ISSN: 0010-2628

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topJaftha, Jacob J.. "The conjugate of a product of linear relations." Commentationes Mathematicae Universitatis Carolinae 47.2 (2006): 265-273. <http://eudml.org/doc/249842>.

@article{Jaftha2006,

abstract = {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 $(ST)^\{\prime \}=T^\{\prime \}S^\{\prime \}$. 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.},

author = {Jaftha, Jacob J.},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {linear relations; conjugates; linear operators; linear relation; conjugate; linear operator},

language = {eng},

number = {2},

pages = {265-273},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {The conjugate of a product of linear relations},

url = {http://eudml.org/doc/249842},

volume = {47},

year = {2006},

}

TY - JOUR

AU - Jaftha, Jacob J.

TI - The conjugate of a product of linear relations

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2006

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 47

IS - 2

SP - 265

EP - 273

AB - 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 $(ST)^{\prime }=T^{\prime }S^{\prime }$. 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.

LA - eng

KW - linear relations; conjugates; linear operators; linear relation; conjugate; linear operator

UR - http://eudml.org/doc/249842

ER -

## References

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- Kato T., Perturbation Theory for Linear Operators, Grundlehren, vol. 132, Springer, Berlin, 1966. Zbl0836.47009
- Sz.-Nagy B., Perturbations des transformations linéaires fermées, Acta Sci. Math. Szeged 14 (1951), 125-137. (1951) Zbl0045.21601MR0047254

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