Two positive operators Two positive operators T £ S acting on a Hilbert sace such that S2 does not majorize T2.

******************************

Define T and S as operators on C2 by T(z1,z2) = (z1 , 0) and S(z1,z2) = (2z1+ z2 , z1+z2).
Then
sp(T) = sp( é
ê
ë
1
0
0
0
ù
ú
û
) = {0,1} Í Â ³ 0, T* = T, sp(S) = sp( é
ê
ë
2
1
1
1
ù
ú
û
) = {
3 +

Ö5

2
} Í Â ³ 0,

S* = S, sp(S-T) = sp( é
ê
ë
1
1
1
1
ù
ú
û
) = {0,2} Í Â ³ 0.
Hence 0 £ T £ S. But sp(S2 - T2) = sp([
4
3
3
2
]) = {3[(+)/] Ö[10] } is not a subset of  ³ 0. Therefore S2 doesn't majorize T2.


File translated from TEX by TTH, version 2.70.
On 22 Feb 2001, 00:22.