Adjoint for operators in Banach spaces
HTML articles powered by AMS MathViewer
- by Tepper L. Gill, Sudeshna Basu, Woodford W. Zachary and V. Steadman PDF
- Proc. Amer. Math. Soc. 132 (2004), 1429-1434 Request permission
Abstract:
In this paper we show that a result of Gross and Kuelbs, used to study Gaussian measures on Banach spaces, makes it possible to construct an adjoint for operators on separable Banach spaces. This result is used to extend well-known theorems of von Neumann and Lax. We also partially solve an open problem on the existence of a Markushevich basis with unit norm and prove that all closed densely defined linear operators on a separable Banach space can be approximated by bounded operators. This last result extends a theorem of Kaufman for Hilbert spaces and allows us to define a new metric for closed densely defined linear operators on Banach spaces. As an application, we obtain a generalization of the Yosida approximator for semigroups of operators.References
- Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
- Leonard Gross, Abstract Wiener spaces, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 31–42. MR 0212152
- William E. Kaufman, A stronger metric for closed operators in Hilbert space, Proc. Amer. Math. Soc. 90 (1984), no. 1, 83–87. MR 722420, DOI 10.1090/S0002-9939-1984-0722420-9
- J. Kuelbs, Gaussian measures on a Banach space, J. Functional Analysis 5 (1970), 354–367. MR 0260010, DOI 10.1016/0022-1236(70)90014-5
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
- A. Pełczyński, All separable Banach spaces admit for every $\varepsilon >0$ fundamental total and bounded by $1+\varepsilon$ biorthogonal sequences, Studia Math. 55 (1976), no. 3, 295–304. MR 425587, DOI 10.4064/sm-55-3-295-304
- J. von Neumann, Uber adjungierte Funktionaloperatoren, Annals of Mathematics 33 (1932), 294–310.
Additional Information
- Tepper L. Gill
- Affiliation: Department of Electrical Engineering, Howard University, Washington, DC 20059
- Email: tgill@howard.edu
- Sudeshna Basu
- Affiliation: Department of Mathematics, Howard University, Washington, DC 20059
- MR Author ID: 628867
- Email: sbasu@howard.edu
- Woodford W. Zachary
- Affiliation: Department of Electrical Engineering, Howard University, Washington, DC 20059
- Email: wwzachary@earthlink.net
- V. Steadman
- Affiliation: Department of Mathematics, University of the District of Columbia, Washington, DC 20058
- Received by editor(s): May 7, 2002
- Received by editor(s) in revised form: January 8, 2003
- Published electronically: September 22, 2003
- Communicated by: Joseph A. Ball
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 1429-1434
- MSC (2000): Primary 46B99; Secondary 47D03
- DOI: https://doi.org/10.1090/S0002-9939-03-07204-6
- MathSciNet review: 2053349