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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.