Laboratory of Hilbert Spaces
About the Section
The main topics of investigations in recent 15 years were
centered around theory of unbounded hyponormal operators,
unbouded Toeplitz operators and (mostly) unbounded Jacobi
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
- Detailed analysis of various classes of unbounded Teoplitz
operators in the Segal-Baragmann space. Various decompositions
and spectral properties of these operators were found
- 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
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.