By D. Z. Arov (auth.), I. Gohberg, L. A. Sakhnovich (eds.)

This booklet is devoted to the reminiscence of a very good mathematician and character, Vladimir Petrovich Potapov, who made very important contributions to and exerted significant impression within the parts of operator concept, advanced research and their issues of juncture. The booklet commences with insightful biographical fabric, after which offers a set of papers on various points of operator idea and complicated research protecting these fresh achievements of the Odessa-Kharkov tuition within which Potapov was once very lively. The papers take care of interrelated difficulties and strategies. the most themes are the multiplicative constitution of contractive matrix and operator features, operators in areas with indefinite scalar items, inverse difficulties for structures of differential equations, interpolation and approximation difficulties for operator and matrix services. The e-book will attract a large staff of mathematicians and engineers, and masses of the fabric can be utilized for complex classes and seminars.

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As soon as I understood this, I started to solve the problem in the scalar case (n = 1) by considering a generalization of the Nevanlinna-Pick problem. Due to the experience of my previous work on scattering matrices and on continuation problems, it was more natural to consider instead of P n the class Bn = Bnxn' where Bnxm is the class of holomorphic contraction n x m matrix functions S(A) defined for Re A > 0, and Z = (I - S)(I + S)-I. For S E Bnxm, the D-representation corresponding to (1) is written in the form (2) where f = const.

Krein on my scientific work new parametrization of the rational matrix function Z(A) of the class P n in the lossy case; it is an alternative to the known additive Nevanlinna formula for Z(A)( E Pn)' In distinction from the latter, this parametrization of Z(A) is defined by a finite number of the zeros Ak of the matrix function Z(A) + Z*( -~), the poles of A(A), and parameters - the matrix parameters of the elementary multipliers for A( A) in (1). 4. DARLINGTON METHOD IN THE GENERAL THEORY OF PASSIVE SYSTEMS V.

G. Sinai (he sent manuscripts to me), participation in Rohlin's seminar (I went on a mission to Leningrad for consultation with V. A. Rohlin), and interaction with V. A. Rohlin, L. M. Abramov, A. M. Z. Arov 4 and other participants in Rohlin's seminar. I saw that the specialists were interested in my investigations. 2. A TILT TOWARD OPERATOR THEORY At the same time that I worked on my dissertation, I attended KreIn's lectures on the spectral theory of the string and other differential operators and actively participated in KreIn's seminar.

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