C*-algebras Volume 5: Selected Topics by Corneliu Constantinescu

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By Corneliu Constantinescu

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2) (0n)new is a decreasing family in JR+\{0}. 3) (Xn)neY and (Yn)ncg are orthonormal families in E and F , respectively. 4) u = E e~(-I~)y~. 11, N = N ( u ) and for every n 9 N . 1 a), every compact operator admits a Schatten decomposition. 5 ( 0 ) If u 9 K:(E,F) and v 9 s then there are orthonormal bases (x~)~ei, (Y~)~L of E and F , respectively, such that n 9 I ML :~ (vuxnixn) = (uvYniYn), 9 I \ L ====v<~x~lx~>)~ e L \ I ~ o, ( ~ v y ~ l y ~ ) = o. l=,~)yn nEN be a Schatten decomposition of u and let (x~)~ei, (Y~)~eL be orthonormal bases of E and F , respectively, such that N=INL.

7). ~a~)t <_ Ilx~ll I I t ~ . ~ a m l l < 1 nEN for every m E M and 38 6. 10 c)). 11, w E/~I(E) and I1~11, < Ilull~llvllq. 10 c) ). 22 ( 0 ) Take u,v E f_,(E, F) with uov*=O a) (Keru) • C Kerv, b) [[u § v i i - sup(llull, Ilvll). and v* o u = O . (Keru*) • C Kerr*, Imu 2_ I m v . Ix,)y, , f g = o. tel If p 9 [1, c~[ and u, v 9 s F) , then I1~ + vll~ = Ilull~ + t1~11~. 4) and (Keru) • (Kerv) • Imu/Imv. b) We may assume that fl~l < I1~11. Put G "- Ker u. By a), I1~11 = IIw~zll = I1(~ + ~ ) ~ x l l < I1~ + vii I ~ x l l _< i1~ + vii IIx I for every x E E , so that By a), II(u + v)xll 2 = Ilux + vxll 2 = Ilu~clx + w ~ x l l 2 - - = I l u ~ l x l l 2 + Ilwcxll 2 < Ilull211~cxll 2 + llvll211~xll 2 < 4o 6.

TI~=Tn0 There is, furthermore, an no C IN such that 2too Z cp 0m(~. -- u)~ < 7 ' m=l for every n E IN, with n _> no. e. lim Ilu~ - u[I p = O. 1 s Theorem 37 ( 0 ) Let p,q be conjugate exponents. q(F,G). Then v o u E Et(E,G) and (HOlder inequality). 13 b), p, q E [1, c~[. we may assume that E Ozm<'lam)X~ ' E mEM Let fin <'lyn)bn mEN by Schatten decompositions of u and v, respectively. 10, there is a t E s G) such that Ker(vou)-Kert, vou-tow. Let A be an orthonormal basis of E and take B E ~ s ( A ) .

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