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If we let $H$ be a Hilbert space with inner product $\langle.,.\rangle$. And we fix $y, z \in H$. Then let $T:H\rightarrow H$ be the bounded linear operator $Tx = \langle x,y\rangle z$. Then what is the Hilbert adjoint operator $T^{\star}: H \rightarrow H$ such that $\langle Tx,w\rangle = \langle x, T^\star w\rangle$ for all $x,w \in H$.

I have started the argument along the following lines

$$\langle Tx, w \rangle = \langle\langle x, y \rangle z, w \rangle = \langle x, y\rangle \langle z, w \rangle =...$$

But I can't see where to argue next. I would appreciate any help! Thanks :)

Also are there any general methods for computing adjoint operators?

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Note that $$ \langle Tx, w \rangle = \langle\langle x, y \rangle z, w \rangle = \langle x, y\rangle \langle z, w \rangle = \langle x, \overline{\langle z, w \rangle} y\rangle = \langle x, \langle w, z \rangle y\rangle = \langle x, T^*w \rangle $$ so $$ T^*w=\langle w, z \rangle y $$

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  • $\begingroup$ Of course! Thanks for that. Very much appreciated! $\endgroup$ – Tim Apr 27 '14 at 17:00

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