An involution An involution # on Banach algebra M4(C), two normal matrix T and S such that TS = ST but TS# ¹ S#T, S+T isn't normal and ||SS#|| ¹ ||S||2.

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Set
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ø 		,      S = æ
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ø 		        and        T = æ
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ø 		.
Then Q# = U-1Q*U where Q* denote the conjugate transpose of Q is an involution on M4(C). An straightforward computation shows that S and T has desired properties.
Ref.
[Rud1] W. Rudin, Functional analisis, McGraw-Hill, 1989.

File translated from TEX by TTH, version 2.70.
On 22 Feb 2001, 00:18.