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I am trying to solve the following problem:

$T\in B(X)$. If each $\lambda\in \sigma(T)\backslash\{0\}$ is an isolated point in $\sigma(T)$ and the riesz projection corresponding to $\lambda$ is of finite rank, then $T$ is of the form $Q+K$, where $Q$ is a quasinilpotent operator and $K$ is compact.

I manage to show that the essential spectrum of $T$, $\sigma_e(T)=\{0\}$, so $T$ is essentially quasinilpotent, or, in other words, it is quasinilpotent in the algebra $B(X)/\mathcal{K}(X)$. But this does not imply the wanted result unless we can find an quasinilpotent operator in the class of $T$ in the calkin algebra.

I wonder whether someone has a hint on how to do this.

Or even better, is there some general technique that lifts information from $B(X)/\mathcal{K}(X)$ to $B(X)$? Calkin algebra would not be so useful if one could not find an effective way to do this right?

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up vote 0 down vote accepted

The question concerning the quoted problem is solved by West. He uses a technique called super-diagonalization.

In general, the problem of lifting from $B(X)/\mathcal{K}(X)$ does not have a solution as far as I know.

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