# Show that $T$ has an adjoint, and describe $T^*$ explicitly.

Let $$V$$ be an inner product space and $$\beta, \gamma$$ fixed vectors in $$V$$. Show that $$T \alpha = (\alpha\mid\beta) \gamma$$ defines a linear operator on $$V$$. Show that $$T$$ has an adjoint, and describe $$T^*$$ explicitly.

Attempt:

The Linearity is easy. The problem is to find $$T^*$$.

$$(Tv\mid u)=((v\mid \beta) \gamma\mid u)= (v\mid\beta)(\gamma\mid u)=(v\mid\overline{(\gamma\mid u)}\beta)$$.

Then $$T ^ * u = \overline{( \gamma\mid u)} \beta = {(u\mid\gamma)}\beta$$.

Is correct? I'm not sure you can do this $$((v\mid\beta)\gamma\mid u)= (v\mid\beta)(\gamma\mid u)$$. Any suggestion?

• That's perfectly fine, as $(v \mid \beta)$ is a scalar like any other. The answer is correct. – Theo Bendit Feb 17 at 13:25
• @TheoBendit Thanks for the comment! – Ricardo Freire Feb 17 at 13:27