Finding the Adjoint of an Operator on a Hilbert Space

[pyf]

Homework Statement

Let H be a Hilbert space, with a in H and b in H.

Let u be an operator on H with u(x)=<b,x>a

Find the adjoint of u.

Thanks!

Homework Equations

<ux,y>=<x,u*y>

The Attempt at a Solution

<ux,y>=<<b,x>a,y> = <b,x><a,y> = <b<a,y>,x>

So it seems that all I need to do is flip that last inner product round. But that will introduce a complex conjugate, stopping me from getting the equation to the right form (maybe?).

 

[Dick]

Is <<b,x>a,y> = <b,x><a,y> really correct?

 

[pyf]

The inner product is linear in its first argument, so I think so.

 

[Dick]

The inner product is linear in its first argument, so I think so.

Ok, I'm used to the convention where it's linear in the second argument and antilinear in the first. Are you sure?

 

[pyf]

I see. Well, this is the form that we've been given. Any ideas?

 

[Dick]

I see. Well, this is the form that we've been given. Any ideas?

Check your definitions again. I have that for a constant c, <x,cy>=c <x,y> and <cx,y>=c* <x,y>. If you take that sort of an inner product the problem works out nicely. If you take the opposite convention then u(cx)=<b,cx>a=(c*)(<b,x>a)=(c*)u(x). That means u ISN'T a linear operator. It's antilinear. The definition of 'adjoint' for antilinear operators needs to be modified. If you REALLY have a backwards convention for the inner product and a forwards convention for the definition of adjoint, then that's messed up.