Weak compactness in the space of operator valued measures M b a ( Σ , ( X , Y ) ) and its applications

N.U. Ahmed

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

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

Abstract

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In this note we present necessary and sufficient conditions characterizing conditionally weakly compact sets in the space of (bounded linear) operator valued measures M b a ( 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 b a ( Σ , ₁ ( X , Y ) ) . This result has interesting applications in optimization and control theory as illustrated by several examples.

How to cite

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N.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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  1. [1] J. Diestel and J.J. Uhl Jr, Vector Measures, American Mathematical Society, Providence, Rhode Island, 1977. 
  2. [2] N. Dunford and J.T. Schwartz, Linear Operators, Part 1, General Theory, Second Printing, 1964. 
  3. [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. [4] T. Kuo, Weak convergence of vector measures on F-spaces, Math. Z. 143 (1975), 175-180. doi: 10.7151/dmdico.1136 
  5. [5] I. Dobrakov, On integration in Banach spaces I, Czechoslov Math. J. 20 (95) (1970), 511-536. Zbl0215.20103
  6. [6] I. Dobrakov, On integration in Banach spaces IV, Czechoslov Math. J. 30 (105) (1980), 259-279. Zbl0452.28006
  7. [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. [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. [9] N.U. Ahmed, Impulsive perturbation of C₀-semigroups by operator valued measures, Nonlinear Funct. Anal. & Appl. 9 (1) (2004), 127-147. 
  10. [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. [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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