By Daniel Alpay, Victor Vinnikov

This quantity provides the refereed court cases of the convention in Operator The ory in Honour of Moshe Livsic eightieth Birthday, held June 29 to July four, 1997, on the Ben-Gurion collage of the Negev (Beer-Sheva, Israel) and on the Weizmann In stitute of technology (Rehovot, Israel). the amount includes papers in operator thought and its purposes (understood in a really large sense), a lot of them reflecting, 1 without delay or in a roundabout way, a profound influence of the paintings of Moshe Livsic. Moshe (Mikhail Samuilovich) Livsic used to be born on July four, 1917, within the small city of Pokotilova close to Uman, within the province of Kiev within the Ukraine; his family members moved to Odessa while he used to be 4 years previous. In 1933 he enrolled within the division of Physics and arithmetic on the Odessa nation collage, the place he turned a pupil of M. G. Krein and an lively player in Krein's seminar - one of many centres the place the guidelines and techniques of practical research and operator conception have been being constructed. along with M. G. Krein, M. S. Livsic was once strongly prompted B. Va. Levin, an excellent expert within the thought of analytic services. A through deep figuring out of operator thought in addition to functionality concept and a penetrating seek of connections among the 2, have been to turn into one of many landmarks of M. S. Livsic's paintings. M. S. Livsic defended his Ph. D.

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Those lawsuits comprise lectures given on the N. A. T. O. complicated examine Institute entitled "Scattering conception in arithmetic and Physics" held in Denver, Colorado, June 11-29, 1973. we've got assembled the most sequence of lectures and a few awarded via different members that appeared evidently to counterpoint them.

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**Example text**

And . d'P* dx UI U>. Z + A 2'P * UI u>. - A I'P * U2 U >. = 'P * "( in u>.. Adding we obtain (i d~ + A2(X))'P*(X)(JIU>. I)'P*(X)(J2U>.. (X) = (Al(X) - AI)(i d~ + A2(x))f,\(x). (x) of the equation (42) satisfies also the equation (43). 0 Theorem 8 shows that there exist input-output relations between differential equations with spectral parameter A related to the given vessel field with zero curvature. Definition. The differential operator iO'l dv + bout(x) + A0'2]V dx is said to be a vascular transformation of the differential operator .

In this paper we use, in a slightly different form, some of his results. References [1] H. Gauchman, Operator colligations on differentiable manifolds, Toeplitz Centennial (Tel Aviv, 1981), edited by I. Gohberg, Operator Theory Adv. Appl. 4, Birkhiiuser, Basel, 1982, pp. 271-302. [2] H. Gauchman, On nonselfadjoint representations of Lie algebras, Integral Equations Operator Theory 6 (1983), 672-705. [3] M. S. Livsic, N. Kravitsky, A. S. Markus, V. Vinnikov, Theory of Commuting Nonselfadjoint Operators, Mathematics and its Applications 332, Kluwer Academic Publishers Group, Dordrecht, 1995.

Livsic Using the linkage condition we obtain From f) ( (J2 = ii - i