King Saud University
  Help (new window)

تحميل الدليل التدريبي

أسئلة شائعة


Functional Analysis II

Operator Theory, Spectral Theory and Banach Algebras



I. Preliminaries

II. Spectral theory of bounded linear operators on Banach spaces

  • Auxiliary material
  • The spectrum
  • Functional calculus
  • Spectra of projections and compact operators

III. Spectral  Theory in Hilbert spaces

  • Basics
  • The spectrum and the numerical range
  • Spectra of unitary and normal operators
  • Hilbert-Schmidt operators (and more Schatten class operators)
  • The Spectral theorem for compact normal operators
  • Positive operators and polar decomposition
  • The spectral theorem for bounded self-adjoint operators

IV. Banach algebras

  • Generalities on Banach algebras
  • Spectral theory in Banach algebras
  • Gelfand theory (Gelfand topology, Gelfand transform and Structure space)
  • Applications to Fourier analysis

V. C*-algebras and spectral theory of normal operators

  • C*-algebras
  • Gelfand-Naimark theorem
  • The spectral theorem
  • Spectral measure and spectral decomposition of normal operators


Main references used in the course:

  • Douglas, R.G.:  Banach algebra techniques in operator theory. 2nd edition, Springer. Academic Press 1972.
  • Halmos, P.: A Hilbert space problem book. 2nd Ed., Graduate Texts in Mathematics, 19. Encyclopedia of Math. and its Appl., 17. Springer-Verlag, New York-Berlin, 1982.
  • Murphy, G.J.:  C*-algebras and operator theory. Academic Press, 1990.
  • Rudin, W.: Functional analysis. McGraw-Hill 1973.  


Samples of previous exams and other course material: (more is coming soon)

Examples of midtem exams: 582--exam2007.pdf




Examples of final exams:    582---final1428.pdf





King   Saud University. All rights reserved, 2007 | Disclaimer | CiteSeerx