By Leonid L. Waksman
Class of commuting non-selfadjoint operators is likely one of the so much hard difficulties in operator idea even within the finite-dimensional case. The spectral research of dissipative operators has ended in a sequence of deep leads to the framework of unitary dilations and attribute operator services. It has became out that the idea needs to be in response to analytic services on algebraic manifolds and never on features of numerous autonomous 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 elements are attached within the wide-spread case, through a definite algebraic equation whose measure doesn't exceed the measurement of the sum of the levels of imaginary parts." Such investigations were performed in instructions. considered one of them, offered through L.L.Waksman, is said to semigroups of projections of multiplication operators on Riemann surfaces. one other path, that's awarded the following by means of M.S.Livsic relies on operator colligations and collective motions of structures. each given wave equation may be received as an exterior manifestation of collective motions. The algebraic equation pointed out above is the corresponding dispersion legislations of the input-output waves.
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