Download Completeness of Root Functions of Regular Differential by Sasun Yakubov PDF

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}.

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