# Approximation of Hilbert Space Operators (Research Notes in by Domingo A. Herrero

By Domingo A. Herrero

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It is clear that l/2c ~ n > l/2c-l. Hence, {n+l) > l/2c and n-1 trace {A) ~ {n+l) - 2c}:j=O j = (n+l)- {n+l) nc = {n+l) {1-nc) > l/4c. 10) IIA-HII ~ I<{A-H)x,x>l > a-(a-c) =c, a contradiction. Hence, rank F{[a-c,aa)) ~rank E{[a,aa)) and, by symmetry, rank F([a,ao)) ~rank E{[a-c,ao))' for all a E [0,1]. It folows that 0 = trace (H) > trace {A) - kc > 1/ 4c - kc. Hence, c > 1/2/k. m1 {A-A·)kj {A. m1 k. =d and Tis sim J= J l. J J= J ilar to the Jordan form ej:1 {Aj+qk·). n 1 h. • =JJk =A 1, J l. J= 1 l.

In this case, Rtx E ker(A. 1-T) and since the last space is one-dimensional, ker ( A. 1 -T) c ran Rt. This contradicts ran Rt and Pker(Al-T)y f 0. -A 0 )Rt+l is invertible for all A. -Ao)Rt+l]-1 is a right resolvent forT in n 1 \S 1 • This completes the proof. 18. -T) = n, A. E Qn. Foro any £ > 0, there exists a right resolvent F ofT on Qn exaept foro an at most denumerable set Sn, whiah does not accumulate on nn, and satisfies sn c cannJ£. PROOF. We proceed by induction on n. -T) -l, A. E Qn) and the case n = 1 is contained in the preceding lemma.

It is completely apparent that ~ defines a bounded linear mapping from L[H) into itself. l'le shall verify that~ is a right inverse for TAB-~· Indeed, . , 21wi = 1 on L [H). Hence, ar (TAB) c ar (A) -a R. (B) and the proof of (i) is complete. 2, aw(TAB) c aR. (A)- ar(B) and aw(TAB) c aR. (TAB). (A)and some f3 "ar(B). This means that, given € > 0, we can find unit vectors x, yin H such that II