Operator Theory on Hardy Spaces

Code: MA648 | L-T-P-C: 3-0-0-6

Prerequisites: MA543 or equivalent

Course Content/ Syllabus: Review of l² and L² -spaces, Hardy spaces: H¹, H² and H^∞; kernel function, reproducing kernel Hilbert spaces, examples of reproducing kernel Hilbert spaces different from Hardy spaces; shift operators (unilateral and bilateral ) on the Hardy spaces, spectrum of shift operators; isometries, Wold-decomposition theorem; inner-outer functions, the inner-outer factorization of functions in H² , Beurling's theorem, The F. Riesz and M. Riesz theorem; Toeplitz matrices, basic properties of Toeplitz operators; introduction to vector-valued Hardy spaces.

Texts:

1. P. R. Halmos, A Hilbert Space Problem Book, Graduate Texts in Mathematics, New-York: Springer Verlag, 1982.
2. J. B. Conway, A course in Operator Theory, Graduate Texts in Mathematics, Volume 21, American Mathematical Society, Providence, RI, 2000.

References:

1. R. A. Martinez-Avendano and P. Rosenthal, An introduction to operators on the Hardy-Hilbert space, Graduate Texts in Mathematics, Springer, New-York, 2007.
2. H. Radjavi and P. Rosenthal, Invariant Subspaces, Second Edition, Springer-Verlag, New-York,1973.
3. R. G. Douglas, Banach Algebra Techniques in Operator Theory, Second Edition, Graduate Texts in Mathematics, Volume 179, Springer-Verlag, New-York, 1998.