The main topics of investigations in recent 15 years were centered around theory of unbounded hyponormal operators, unbouded Toeplitz operators and (mostly) unbounded Jacobi matrices.

The main results are the following:

- Description of basic properties of unbounded hyponormal operators. In particular a characterization of hyponormal generators of $ C-0$ semigroups and generators of cosine functions.(Janas)
- Detailed analysis of various classes of unbounded Teoplitz operators in the Segal-Baragmann space. Various decompositions and spectral properties of these operators were found (Janas-Stochel).
- Precise analysis of asymptotic behaviour of some bases of solutions of the second oredr linear diffrence equations were proved (Janas-Naboko and Janas-Moszynski). Among the most interesting is formulation of the asymptotic formulas for the above bases in the case of double root situation. this was previously known practically only for rational coefficients(Janas).All these asymptotic formulkas found interesting applications in delicate spectral analysis of unbounded (or not) Jacobi operators.

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.

- 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.