I find there is a question for an orthonormal basis. I have a question when the basis is not orthonormal.

Precisely, suppose {ϕi}\{\phi_i\} is a basis, i.e., finite linear combination of ϕi\phi_i is dense, of a separable Hilbert Space H\mathcal{H}. {ϕi}\{\phi_i\} is not necessarily orthonormal.

A bounded operator TT is diagonalizable with respect to {ϕi}\{\phi_i\}, i.e., T(ϕi)=λiϕi,∀i.T(\phi_i)=\lambda_i \phi_i, \forall i.

Then is it true that TT is compact iff λi→0\lambda_i \rightarrow 0 ?

I am having trouble to see a clear relation between an arbitrary basis and the inner product in H\mathcal{H} and could not utilize the result for orthonormal basis.

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