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