Suppose S,TL(V) are self-adjoint. Prove that ST is self-adjoint if and only if ST=TS.
Given S,T L(V) are self-adjoint.
If suppose ST = TS,
ST + TS = 2(ST). Since 2(ST) is self-adjoint, where 2 is a real number.
consider, 2 ST(v), w = 2ST(v), w = v, 2ST(w) = 2 v, ST(w)
therefore, we get,
ST(v), w = v, ST(w) ST is self-adjoint.
Now suppose ST is self-adjoint. then ST(v), w = v, ST(w)
and ST(v), w = v,(ST)*w = v, T*S*w = v, TS(w) (since T, S are self-adjoint)
we get, v, ST(w) = v, TS(w) for all v, w ∈ V
v, (ST-TS)(w) = 0 for all v, w ∈ V ,
so setting v = (ST − T S)w
(ST-TS)(w) , (ST-TS)(w) = 0
||(ST − T S)w||2 = 0 for all w ∈ V ,
therefore,
ST − T S = 0 ST = TS
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