The volume contains selected papers of the Spectral Function Theory
seminar, Leningrad Branch of Steklov Mathematical Institute. The
papers are mostly devoted to the theory of Toeplitz and model
operators. These subjects are considered here from various points
of view. Several papers concern the relationships of Toeplitz
operators to weighted polynomial approximation. Namely, two papers
by B. Solomyak and A. Volberg intensively treat the problem of
spectra! multiplicity f~r analytic Toeplitz operators (which are,
in fact, multiplication operators) and my paper can serve as an
introduction to the problem. This theme of multiplicities is
continued in a paper by V. Vasyunin where the multiplicity of the
spectrum is computed for Hilbert space contractions with finite
defect indices. V. Peller's paper deals with a perturbation theory
problem for Toeplitz operators. In a paper by D. Yakubovich a new
similarity model for a class of Toeplitz operators is constructed.
S. Treil' presents a survey of a part of spectral function theory
for vector valued function (Szego-Kolmogorov extreme prob!ems for
operator weights, bases of vector rational functions, estimations
of Hilbert transform with respect to operator weights, the operator
corona problem). As a concluding remark I dare only note that the
whole collection convinces us once more without a doubt of the
fruitfullness of the natural union of operator theory and complex
analysis (if at all the union of these fields is at all different
from their intersection).
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