- #1

fluidistic

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## Homework Statement

I must show several properties about linear operators

**using the definition of the adjoint operator**.

A and B are linear operator and ##\alpha## is a complex number.

The first relation I must show is ##(\alpha A + B)^*=\overline \alpha A^*+B^*##.

## Homework Equations

The definition I have an an adjoint is: ##A^*## is the adjoint of ##A## if ##\langle g,Af \rangle = \langle A^* g ,f \rangle## where f and g are any vectors in a Hilbert space.

## The Attempt at a Solution

Let ##C^*=(\alpha A+B)^*##. Using the definition of adjoint I get: ##\langle C^*g,f \rangle=\langle g, Cf \rangle \Rightarrow \langle (\alpha A+B)^*g ,f \rangle =#### \langle g, (\alpha A+B)f \rangle =\langle g, \alpha A \cdot f \rangle + \langle g, Bf \rangle = \alpha \langle g, Af \rangle + \langle g ,Bf \rangle = \alpha \langle A^*g, f \rangle + \langle B^*g ,f \rangle##.

But I'm getting lost. I've no idea how I can obtain A, B, A^* and B^* using the definition of the adjoint.

Oh wait, on my draft I think I have finished the "proof". The last expression is worth ##\langle \overline \alpha A^*g ,f \rangle + \langle B^* g, f \rangle = \langle (\overline \alpha A^* + B^*)g ,f \rangle##. Then by associativity ##(\alpha A+B)^*=\overline \alpha A^* + B^*##.

Does this look right?