Hilbert Spaces

Hilbert Spaces

The Section was founded in the 70-ties with Professor Włodzimierz Mlak as head. The main results obtained in the 70-ties and 80-ties are: a general result on partition of spectral sets (now known as the Lautzenheiser-Mlak theorem) (W. Mlak), an operator version of the von Neumann inequality (called also the Arveson-Mlak-Parrot inequality) (W. Mlak), a complete description of circular operators (W. Mlak), canonical decompositions of general operator valued functions (W. Szymański), a characterization of C^*-algebras generated by Toeplitz operators (J. Janas), models for subnormal operators with infinitely connected complement of the spectrum (K. Rudol).

In the 90-ties interesting results were found in the theory of unbounded Toeplitz operators in the Segal-Bargmann space (J. Janas) and in applications of bundle shift models for subnormal operators to boundary values of holomorphic functions (K. Rudol).

In recent years spectral analysis of Jacobi matrices is the main field of investigation. J. Janas (in collaboration with S. Naboko) found some new and interesting theorems in this area.

Selected publications in the 90-ties

J. Janas, An extension of Berezin's approximation method, J. Operator Theory 29 (1993), 43-56.
J. Janas and J. Stochel, Unbounded Toeplitz operators in the Segal-Bargmann space II, J. Funct. Anal. 126 (1994), 418-446.
J. Janas, An operator version of the Newman-Shapiro isometry theorem, Integral Equations Operator Theory 26 (1996), 188-201.
K. Rudol, Spectra of subnormal Hardy type operators, Ann. Polon. Math. 45 (1997), 213-222.