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.

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

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 $\langle XYv,w\rangle$.


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