Here’s why I needed to know about adjoints. In the last wavelets seminar, Papadakis proved the following:

Suppose is a Hilbert space, , . Define the linear mapping , . Then:

1. is a Bessel family iff is bounded
2. is a frame for iff is closed iff iff is invertible iff is invertible iff is invertible

I’m was having a little trouble with the first one, because it involved adjoints in a way that made the proof look incomplete to me (he defined another operator and proved its adjoint was bounded; to everyone else but me, it was clear the adjoint of this operator was in fact ) . I’m currently going over the second one, because I didn’t understand how the proof given.

Hopefully soon, I’ll have these notes up, and I’ll post a link here. They are pretty interesting, for people into signal processing or functional analysis, and easy to follow if you take a few theorems and results for granted. I’d say someone who’s read the multilinear analysis chapter of Rudin and has some minimal experience with using Hilbert space reasoning, could follow this easily. At least after I’m finished rewriting it at my level.

Possibly relevant posts:

• Adjoints (in Hilbert spaces) (10/5/2005)
• Every finite spanning set is a frame (5/26/2006)
• The Gelfand representation theorem (7/4/2008)