Completeness of Root Functions of Regular Differential by Sasun Yakubov

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By Sasun Yakubov

The best mathematical research of assorted normal phenomena is an previous and tough challenge. This publication is the 1st to deal systematically with the final non-selfadjoint difficulties in mechanics and physics. It offers commonly with bounded domain names with tender barriers, but in addition considers elliptic boundary price difficulties in tube domain names, i.e. in non-smooth domain names. This quantity might be of specific worth to these operating in differential equations, sensible research, and equations of mathematical physics.

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6). 3. In terp o latio n spaces. ,• is continuous on (0, 00) in t and the following estimate holds: min{l,<}||w||£;,+f;, < K (t,u ) < max{l,i}||u|l£;,+f;,. An interpolation space for {iJoj-Ei} by the /f-method is defined as follows: (Eo,Ei)0,p — {u| u € Eo + E i, ||w||(£o,Bi )*,i. 1/ p < 00, 0 < f l < l , l < p < 00}, 1 . 7) L em m a 2,3. Let Eja be a subspace of Ej, j = 0, 1. Then the embedding (Eoo,Eio)e,p C (Eo,Ei)e,p, 0 < 0 < 1 , l^ p < o o , is continuous. P roof. B, Jo The case p = oo is proved in a similar way.

10) we obtain A*Auj = sj(A-,H,Hi)uj, AA*Vj = sj{A;H,Hi)vj. 9) follows. 1. 8. ^B -,H i,H i) < ^B'^b (Hi ,h )S){A\H,H i ), ( 1. 11) ( 1. 12) P ro o f. , A*B*BA < Hence, \^{A-B*B A) < \\B\\%^^^^j,^Xj{A*A). 13) On the other hand, by the definition s)(BA; H, H) = \j{A*B*BA), H, Hi ) = \j{A*A). 11). 12) are also proved. B). 1*10. WeyPs inequality: a co rrelation betw een th e eigenvalues and th e s-num bers. , Sj = Sj(A) = Sj{A\H,H). 36 2. 10. 3,1]. Let A in H be compact. Then for any system of the elements Uk, = the following correlation is valid: det((Awj, < s l s l - - - s ^ det((uj, Uk))i.

An element u E H is uniquely representable in the form u = Ui + where u\ E Hi^ E H ^, The operator Pu = U\ is called an operator of orthogonal projection onto Hi and has the following prop­ erties: P is bounded in if, = P^ P* = P , ||P|| = Pu = u E H \. 3. L aurent series expansion of th e resolvent. 2) where Ajb, fc = —g , . . 2) converges by the operator norm in E. 2. Let Aq be an eigenvalue of the operator A and a pole of the £nite order of the resolvent R(X,A). p(^A —• Aq/ ) C ( A XqI^A—p = A_(p_|_i), p = (3) Ao(A - Ao/) C (A - Ao/)Ao = A_i -f I; (4) A -,(A - Ao/) C (A - Ao/)A_, = 0.

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