By Sasun Yakubov

The best mathematical research of assorted typical phenomena is an previous and hard challenge. This publication is the 1st to deal systematically with the final non-selfadjoint difficulties in mechanics and physics. It bargains often with bounded domain names with tender obstacles, but additionally considers elliptic boundary worth difficulties in tube domain names, i.e. in non-smooth domain names. This quantity may be of specific price to these operating in differential equations, practical research, and equations of mathematical physics.

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7(A). 11. Classes By (7p{H,Hi), p > 0, we denote the space of operators A that act from H into H\ compactly and for which \\A\\p = J2^^ j { A -,H ,H ,)< c<>. 1. COMPLETENESS FOR AN OPERATOR FROM ( T j, { H ) 37 For brevity denote (7p{H^H) = cTp{H), Let us show that (Tp{H) under p > 1 is a linear manifold. Let A 6 (Tp{H)^ B G ap{H). Represent the natural number m uniquely in the form m= fc = 0, . . , oo, ^ = 0, 1. , oo. Hence, i/p ||i4 + B\\p - I ^ s ’^{A + B ) i/p < + < 2»/'>(||A||p + I l S y .

Consider the following function F{X) = {{I - X B ) - \ , u ) , A^ = A*‘(B). Show that points A¡^^(J5) are removable singular points of F {\). (A-A ,-‘ )”(B„^,n). 3 we have BnV E N\^ C (spB)_o, n = —(/jfc,. . , —1. Hence, the main part of the Laurent series expansion for F(X) at the point Aj^^ is equal to zero. Denote the restriction of B* onto (spB)io by B*, Since u G (spB)io) then F{\ ) = (u, ( / - XB*)-^u) = {v, ( / - XB^y^u). :o by Q, Then on (spB)fo we have B* = B*Q and F{X) = { v , { I - XB*Q) - ^ u) .

P roof. B, Jo The case p = oo is proved in a similar way. 96]. 2. INTERPOLATION OF SPACES AND OPERATORS 19 There are other methods of interpolation as well. Many of them coincide with interpolation by the iiT-method up to the norm equivalency. But the complex method of interpolation generates new interpolation spaces. Let S denote a stripe S = {z\ z £ 0 < Rear < 1} of the complex plane, and S is its closure. Let {E q,E i } be an interpolation pair. 56] F{Eo^Ei) = {u{z)\ u(z) is an (jE7o+-®i)-valued function, continuous in 5 and analytic in 5, sup^^^ \\u{z )\\eo-¥Ei < oo, u(j + it) € j = 0,1, t G R, u{j + it) is a continuous as JS^-valued function of t, ||u ||f (Eo,Ei ) = maxj=o,i sup^gj^ \\u(j + iOllfi,- < oo}.