# Eigenvalues of a compact positive definite operator

1. Dec 20, 2012

### SVD

eigenvalues of a compact positive definite operator!!!

Let A be a compact positive definite operator on Hilbert space H.
Let ψ1,...ψn be an orthonormal set in H.
How to show that <Aψ1,ψ1>+...+<Aψn,ψn> ≤ λ1(A)+...+λn(A), where
λ1≥λ2≥λ3≥..... be the eigenvalues of A in decreasing order.
Can someone give me a hint???

2. Dec 20, 2012

### Staff: Mentor

Re: eigenvalues of a compact positive definite operator!!!

Both the left and right expression look like tr(A).

3. Dec 20, 2012

### micromass

Re: eigenvalues of a compact positive definite operator!!!

Try induction.

Do you know that

$$\lambda_1=sup\{<Ax,x>~\vert~x\in H,~\|x\|=1\}$$

??

If you know this, then the case n=1 should be easy. Can you find an argument to deal with the other cases?