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Operators on Banach Space

In: Functional Analysis and the Feynman Operator Calculus

Author

Listed:
  • Tepper L. Gill

    (Howard University, Departments of Electrical and Computer Engineering)

  • Woodford Zachary

    (Howard University, Departments of Electrical and Computer Engineering)

Abstract

The Feynman operator calculus and the Feynman path integral develop naturally on Hilbert space. In this chapter we develop the theory of semigroups of operators, which is the central tool for both. In order to extend the theory to other areas of interest, we begin with a new approach to operator theory on Banach spaces. We first show that the structure of the bounded linear operators on Banach space with an S-basis is much closer to that for the same operators on Hilbert space. We will exploit this new relationship to transfer the theory of semigroups of operators developed for Hilbert spaces to Banach spaces. The results are complete for uniformly convex Banach spaces, so we restrict our presentation to that case, with one exception. In the Appendix (Sect. 5.3), we show that all of the results in Chap. 4 have natural analogues for uniformly convex Banach spaces.

Suggested Citation

  • Tepper L. Gill & Woodford Zachary, 2016. "Operators on Banach Space," Springer Books, in: Functional Analysis and the Feynman Operator Calculus, chapter 0, pages 193-235, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-27595-6_5
    DOI: 10.1007/978-3-319-27595-6_5
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