0
$\begingroup$

Consider the linear and bounded operators $X$ and $Y$on a Hilbert space $\mathcal{H}$ with inner product $\langle \cdot,\cdot \rangle$. How can I show that

$$ \langle XY \boldsymbol{v}, \boldsymbol{w}\rangle = \langle \boldsymbol{v}, Y^*X^*\boldsymbol{w}\rangle, \quad \forall\boldsymbol{v},\boldsymbol{w}\in\mathcal{H}$$

where $^*$ denotes the adjoint operator. I'm not sure how to get startet, so any hint is appreciated.

$\endgroup$
  • $\begingroup$ Do you know the definition of an adjoint operator? $\endgroup$ – Sten Mar 3 '15 at 18:22
1
$\begingroup$

Hint: what does it mean to be an adjoint? By definition, $\langle Xv, w \rangle = \langle v, X^* w\rangle$ for any $v,w \in \mathcal{H}$. Now apply this twice to $\langle XYv,w\rangle$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.