Here's an "exercise" which I thought should be easy, but which I find myself unable to do.

Let $V$ be a Banach space.

Recall that an operator $f:V\to V$ is **trace-class** if it is in the image of the natural map $V\otimes_\pi V'\to \mathcal L(V,V)$, where $\otimes_\pi$ denotes the projective tensor product, and $V'=\mathcal L(V,\mathbb C)$ is the continuous dual of $V$.

Exercise:[Edit: this is probablyFALSE; thank youMateusz Wasilewski]

Let $f:V\to V$ be a trace-class linear map which is "diagonalizable", in the sense that the linear span of its eigenspaces is dense in $V$. Write $\lambda_1,\lambda_2,\ldots$ for the eigenvalues of $f$ (enumerated with multiplicities). Then $$\sum |\lambda_n|<\infty.$$

Given that my previous "exercise" turned out to be probably false, here's a new one:

Exercise v2:

Let $f:V\to V$ be a trace-class linear map which is diagonalizable (in the same sense as above), and whose eigenvalues $\lambda_1,\lambda_2,\ldots$ are all positive real numbers. Then $$\sum \lambda_n<\infty.$$

Can someone please help me prove it?

And if this second exercise also too difficult, here's an even simpler one (which I also don't know how to prove):

Exercise v3:

Let $f:V\to V$ be a trace-class linear map which is diagonalizable (in the same sense as above), with eigenvalues $\lambda_1,\lambda_2,\ldots$. Then $$\lim_{n\to\infty} \lambda_n=0.$$

PS: I'm also interested in the same question when $V$ is Frechet, or when $V$ is a general complete locally convex topological vector space.

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