By Leonid L. Waksman
Class of commuting non-selfadjoint operators is without doubt one of the such a lot tough difficulties in operator thought even within the finite-dimensional case. The spectral research of dissipative operators has resulted in a chain of deep ends up in the framework of unitary dilations and attribute operator features. It has became out that the idea should be in line with analytic services on algebraic manifolds and never on services of a number of self sustaining variables as was once formerly believed. This follows from the generalized Cayley-Hamilton Theorem, as a result of M.S.Livsic: "Two commuting operators with finite dimensional imaginary components are attached within the favourite case, by means of a undeniable algebraic equation whose measure doesn't exceed the measurement of the sum of the levels of imaginary parts." Such investigations were conducted in instructions. one in all them, provided by way of L.L.Waksman, is said to semigroups of projections of multiplication operators on Riemann surfaces. one other course, that is provided right here by means of M.S.Livsic is predicated on operator colligations and collective motions of structures. each given wave equation will be got as an exterior manifestation of collective motions. The algebraic equation pointed out above is the corresponding dispersion legislation of the input-output waves.
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