Question

Suppose S,TL(V) are self-adjoint. Prove that ST is self-adjoint if and only if ST=TS.

Answer 1

Given S,T $\isin$ L(V) are self-adjoint.

If suppose ST = TS,

$\implies$ ST + TS = 2(ST). Since 2(ST) is self-adjoint, where 2 is a real number.

consider, 2$\langle$ ST(v), w $\rangle$ = $\langle$ 2ST(v), w $\rangle$ = $\langle$ v, 2ST(w) $\rangle$ = 2$\langle$ v, ST(w) $\rangle$

therefore, we get,

$\implies$ $\langle$ ST(v), w $\rangle$ = $\langle$ v, ST(w) $\rangle$ $\implies$  ST is self-adjoint. 

Now suppose ST is self-adjoint. then $\langle$ ST(v), w $\rangle$ = $\langle$ v, ST(w) $\rangle$

and $\langle$ ST(v), w $\rangle$ = $\langle$ v,(ST)*w $\rangle$ = $\langle$ v, T*S*w $\rangle$ = $\langle$ v, TS(w) $\rangle$ (since T, S are self-adjoint)

we get, $\langle$ v, ST(w) $\rangle$ = $\langle$ v, TS(w) $\rangle$ for all  v, w ∈ V

$\implies$ $\langle$ v, (ST-TS)(w) $\rangle$ = 0  for all v, w ∈ V ,

so setting v = (ST − T S)w

$\implies$ $\langle$ (ST-TS)(w) , (ST-TS)(w) $\rangle$ = 0

$\implies$ ||(ST − T S)w||² = 0 for all w ∈ V ,

therefore,

ST − T S = 0  $\implies$ ST = TS