# eigenvalues of a compact positive definite operator!!!

by SVD
Tags: compact, definite, eigenvalues, operator, positive
 P: 6 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 +...+ ≤ λ1(A)+...+λn(A), where λ1≥λ2≥λ3≥..... be the eigenvalues of A in decreasing order. Can someone give me a hint???
 Mentor P: 10,769 Both the left and right expression look like tr(A).
 Mentor P: 16,543 Try induction. Do you know that $$\lambda_1=sup\{~\vert~x\in H,~\|x\|=1\}$$ ?? If you know this, then the case n=1 should be easy. Can you find an argument to deal with the other cases?

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