# Weak compactness in the space of operator valued measures ${M}_{b}a(\Sigma ,(X,Y))$ and its applications

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2011)

- Volume: 31, Issue: 2, page 231-247
- ISSN: 1509-9407

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topN.U. Ahmed. "Weak compactness in the space of operator valued measures $M_ba(Σ,(X,Y))$ and its applications." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 31.2 (2011): 231-247. <http://eudml.org/doc/271180>.

@article{N2011,

abstract = {In this note we present necessary and sufficient conditions characterizing conditionally weakly compact sets in the space of (bounded linear) operator valued measures $M_\{ba\}(Σ,(X,Y))$. This generalizes a recent result of the author characterizing conditionally weakly compact subsets of the space of nuclear operator valued measures $M_\{ba\}(Σ,₁(X,Y))$. This result has interesting applications in optimization and control theory as illustrated by several examples.},

author = {N.U. Ahmed},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {space of operator valued measures; weak compactness; semigroups of bounded linear operators; optimal structural control},

language = {eng},

number = {2},

pages = {231-247},

title = {Weak compactness in the space of operator valued measures $M_ba(Σ,(X,Y))$ and its applications},

url = {http://eudml.org/doc/271180},

volume = {31},

year = {2011},

}

TY - JOUR

AU - N.U. Ahmed

TI - Weak compactness in the space of operator valued measures $M_ba(Σ,(X,Y))$ and its applications

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2011

VL - 31

IS - 2

SP - 231

EP - 247

AB - In this note we present necessary and sufficient conditions characterizing conditionally weakly compact sets in the space of (bounded linear) operator valued measures $M_{ba}(Σ,(X,Y))$. This generalizes a recent result of the author characterizing conditionally weakly compact subsets of the space of nuclear operator valued measures $M_{ba}(Σ,₁(X,Y))$. This result has interesting applications in optimization and control theory as illustrated by several examples.

LA - eng

KW - space of operator valued measures; weak compactness; semigroups of bounded linear operators; optimal structural control

UR - http://eudml.org/doc/271180

ER -

## References

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- [3] J.K. Brooks, Weak compactness in the space of vector measures, Bulletin of the American Mathematical Society 78 (2) (1972), 284-287. doi: 10.1090/S0002-9904-1972-12960-4 Zbl0241.28011
- [4] T. Kuo, Weak convergence of vector measures on F-spaces, Math. Z. 143 (1975), 175-180. doi: 10.7151/dmdico.1136
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- [6] I. Dobrakov, On integration in Banach spaces IV, Czechoslov Math. J. 30 (105) (1980), 259-279. Zbl0452.28006
- [7] J.K. Brooks and P.W. Lewis, Linear operators and vector measures, Trans. American Math. Soc. 192 (1974), 139-162. doi: 10.1090/S0002-9947-1974-0338821-5 Zbl0331.46035
- [8] N.U. Ahmed, Vector and operator valued measures as controls for infinite dimensional systems: optimal control Diff. Incl., Control and Optim. 28 (2008), 95-131.
- [9] N.U. Ahmed, Impulsive perturbation of C₀-semigroups by operator valued measures, Nonlinear Funct. Anal. & Appl. 9 (1) (2004), 127-147.
- [10] N.U. Ahmed, Weak compactness in the space of operator valued measures, Publicationes Mathematicae, Debrechen, (PMD) 77 (3-4) (2010), 399-413. Zbl1240.28039
- [11] N.U. Ahmed, Some remarks on the dynamics of impulsive systems in Banach spaces, dynamics of continuous, Discrete and Impulsive Systems 8 (2001), 261-274. Zbl0995.34050

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