By Domingo A. Herrero, Constantin Apostol

ISBN-10: 0273085794

ISBN-13: 9780273085799

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Hence, trace (A) (n+1) (n+l) > > l/2c and j = (n+l)—(n+1)ne = (n+l) (1—ne) — > l/4e. Let A = f A dE and H = 5 A dF (spectral decompositions). 10) a contradiction. rank Hence, rank F([a—c,w)) rank F([ a,")) E( U a—c,"))' for all a 0 = trace (H) Hence, c > E([ct,oo)) and, by symmetry, rank 1/2/k. > trace [0,1]. +q1,). h+2 = an upper triangular a1, = a; 8d = then A 8h1+h2_a2, ... , admits a representation as matrix of the form B1 a12 a13 B2 a23 . aid . 1 that U] a12 a13 . . aid p2 P3 . 0 Moreover, the same result applies to any upper triangular representation of A.

If x < (/2/2)IIxjI ker Q, y i. ker /2. We have 1 = /2/2. = the other On and ker ker Q and y i Il(P_Q)*yiI IP*yII = < Q, then ITPxTI (/2/2)IIyII. , Pz = z) be a unit vector and write z=x+y, x Q. Since it follows that IxIHIylI P IPxIl+flPylj ljPx+PyII a contradiction. Hence, (/2/2) < 1, ,/2/2. 0 It readily follows that 62=12= ,/2/2. This is the only known val 2). ue of 6k or 1k (k If k = 3, 111 P3= 1/3 1 1 1 011 and 111 then P3 = 2/3 0 0 2 = 1 2/3[q3+(q3) ], 000 (P is a rank one projection), Q3 and 11P3—Q3Ilis < 2/3.

1—2nc), trace (A) integral part of (l/2e)). It is clear that where n = n > 1/2c—1. Hence, trace (A) (n+1) (n+l) > > l/2c and j = (n+l)—(n+1)ne = (n+l) (1—ne) — > l/4e. Let A = f A dE and H = 5 A dF (spectral decompositions). 10) a contradiction. rank Hence, rank F([a—c,w)) rank F([ a,")) E( U a—c,"))' for all a 0 = trace (H) Hence, c > E([ct,oo)) and, by symmetry, rank 1/2/k. > trace [0,1]. +q1,). h+2 = an upper triangular a1, = a; 8d = then A 8h1+h2_a2, ... , admits a representation as matrix of the form B1 a12 a13 B2 a23 .

### Approximation of Hilbert Space Operators, Volume I by Domingo A. Herrero, Constantin Apostol

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