# Volume 46, № 1-2, 1994

### Mark Grigor'evich Krein

Ukr. Mat. Zh. - 1994. - 46, № 1-2. - pp. 3–4

### On M. G. Krein's papers in the theory of spaces with an indefinite metric

Azizov T. Ya., Ginsburg Yu. P., Langer H.

Ukr. Mat. Zh. - 1994. - 46, № 1-2. - pp. 5–17

This is a surv ey of the papers by M. G. Krein (and his disciples) devoted to the theory of operators in the spaces with indefinite metric and its applications.

### On time dependent orthogonal polynomials on the unit circle

Ukr. Mat. Zh. - 1994. - 46, № 1-2. - pp. 18–36

Two index formulas for operators defined by infinite band matrices are proved. These results may be interpreted as a generalization of the classical theorem of M. G. Krein for orthogonal polynomials. The proofs are based on dichotomy and nonstationary inertia theory.

### New aspects of Krein's extension theory

Brasche J. F., Koshmanenko V. D., Neidhardt H.

Ukr. Mat. Zh. - 1994. - 46, № 1-2. - pp. 37–54

The extension problem for closed symmetric operators with a gap is studied. A new kind of parametrization of extensions (the so-called Krein model) is developed. The notion of a singular operator plays the key role in our approach. We give the explicit description of extensions and establish the spectral properties of extended operators.

### Theory of self-adjoint extensions of symmetric operators. entire operators and boundary-value problems

Gorbachuk M. L., Gorbachuk V. I.

Ukr. Mat. Zh. - 1994. - 46, № 1-2. - pp. 55–62

This is a brief survey of M. G. Krein's contribution to the theory of self-adjoint extensions of Hermitian operators and to the theory of boundary value problems for differential equations. The further development of these results is also considered.

### On M. G. Krein's contribution to the moment problem

Ukr. Mat. Zh. - 1994. - 46, № 1-2. - pp. 63–75

This is a survey of М. G. Krein's papers devoted to the moment problem.

### M. G. Krein's ideas in the theory of orthogonal polynomials

Ukr. Mat. Zh. - 1994. - 46, № 1-2. - pp. 76–86

We present a survey of M. G. Krein's ideas in tho theory of orthogonal polynomials.

### M. G. Krein's investigations in the theory of entire and meromorphic functions and their further development

Ukr. Mat. Zh. - 1994. - 46, № 1-2. - pp. 87–99

This is a survey of М. G. Krein's results in the theory of entire and meromorphic functions. Their further development is also discussed.

### Szego-Kolmogorov-Krein theorems on weighted trigonometrical approximation and Carleman-type relations

Ukr. Mat. Zh. - 1994. - 46, № 1-2. - pp. 100–127

In view of the well-known Szego - Kolmogorov - Krein theorems on weighted approximation by the functions with semibounded spectrum (on a circle or a line), an efficient construction is suggested, which enables one to realize these approximations. It is based on relations similar to the Carleman tor-mula reconstructing an analytic function in terms its trace on the boundary of the domain of definition.

### On M. G. Krein's work in the theory of linear periodic Hamiltonian systems

Ukr. Mat. Zh. - 1994. - 46, № 1-2. - pp. 128–144

We present the basic ideas of the theory of linear periodic Hamiltonian systems.