Two positive operators
Two positive operators T £ S acting on a Hilbert sace such that S2 does not majorize
T2.
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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
|
|
|
ù ú
û
|
) = {0,1} Í Â ³ 0, T* = T, sp(S) = sp( |
é ê
ë
|
|
|
ù ú
û
|
) = { |
2
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} Í Â ³ 0, |
|
S* = S, sp(S-T) = sp( |
é ê
ë
|
|
|
ù ú
û
|
) = {0,2} Í Â ³ 0. |
|
Hence 0 £ T £ S. But sp(S2 - T2) = sp([
]) = {3[(+)/] Ö[10] } is not a subset of
 ³ 0. Therefore S2 doesn't majorize T2.
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On 22 Feb 2001, 00:22.