- #1

- 45

- 0

## Homework Statement

Prove or give a counterexample: the product of any two self-adjoint operators on a finite-dimensional inner-product space is self-adjoint.

## Homework Equations

The only two equations I've used so far are:

[tex]\left\langle T v, w\right\rangle = \left\langle v, T^{*}w\right\rangle[/tex]

and

[tex]\left(T \circ S \right)^{*} = S^{*} \circ T^{*} [/tex]

## The Attempt at a Solution

I started with the definition of adjoint, using [tex]T \circ S [/tex] :

[tex]\left\langle \left(T \circ S \right) v, w \right\rangle = \left\langle v, \left(T \circ S \right)^{*} w \right\rangle[/tex]

Using the second relevant equation, I got

[tex]\left\langle \left(T \circ S \right) v, w \right\rangle = \left\langle v, \left(S \circ T \right) w \right\rangle[/tex]

So the composition is self-adjoint only if [tex]T[/tex] and [tex]S[/tex] commute. But I don't know where to go from there.

Will self-adjoint operators necessarily commute?