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

• Autorzy: N.U. Ahmed
• Języki publikacji: EN

Abstrakty

EN

In this note we present necessary and sufficient conditions characterizing conditionally weakly compact sets in the space of (bounded linear) operator valued measures $M_{ba}(Σ,𝓛(X,Y))$. This generalizes a recent result of the author characterizing conditionally weakly compact subsets of the space of nuclear operator valued measures $M_{ba}(Σ,𝓛₁(X,Y))$. This result has interesting applications in optimization and control theory as illustrated by several examples.

Słowa kluczowe

EN

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

Kategorie tematyczne

• 46E99: None of the above, but in this section
• 47A55: Perturbation theory
• 46E27: Spaces of measures
• 46B50: Compactness in Banach (or normed) spaces
• 46A50: Compactness in topological linear spaces; angelic spaces, etc.

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae, Differential Inclusions, Control and Optimization
• Rocznik: 2011
• Tom: 31
• Numer: 2
• Strony: 231-247

Daty

wydano

2011

otrzymano

2011-07-25

Twórcy

autor

N.U. Ahmed

• EECS, University of Ottawa, Ottawa, Canada

Bibliografia

• [1] J. Diestel and J.J. Uhl Jr, Vector Measures, American Mathematical Society, Providence, Rhode Island, 1977.
• [2] N. Dunford and J.T. Schwartz, Linear Operators, Part 1, General Theory, Second Printing, 1964.
• [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
• [4] T. Kuo, Weak convergence of vector measures on F-spaces, Math. Z. 143 (1975), 175-180. doi: 10.7151/dmdico.1136
• [5] I. Dobrakov, On integration in Banach spaces I, Czechoslov Math. J. 20 (95) (1970), 511-536.
• [6] I. Dobrakov, On integration in Banach spaces IV, Czechoslov Math. J. 30 (105) (1980), 259-279.
• [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
• [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.
• [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.

Identyfikatory

DOI

10.7151/dmdico.1136