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user473423
user473423

Let $H$ be an infinite-dimensional Hilbert space. Recall that an operator $T$ on $H$ is called quasi-nilpotent, if its spectrum equals $\{0\}$. My questions is this:

Is it possible that the Identity $H\to H$ is a strong limit of quasi-nilpotentnilpotent compact operators in the strong operator topology?

Let $H$ be an infinite-dimensional Hilbert space. Recall that an operator $T$ on $H$ is called quasi-nilpotent, if its spectrum equals $\{0\}$. My questions is this:

Is it possible that the Identity $H\to H$ is a limit of quasi-nilpotent compact operators in the strong operator topology?

Let $H$ be an infinite-dimensional Hilbert space.

Is it possible that the Identity $H\to H$ is a strong limit of nilpotent compact operators?

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user473423
user473423

Strong limits of nilpotent operators

Let $H$ be an infinite-dimensional Hilbert space. Recall that an operator $T$ on $H$ is called quasi-nilpotent, if its spectrum equals $\{0\}$. My questions is this:

Is it possible that the Identity $H\to H$ is a limit of quasi-nilpotent compact operators in the strong operator topology?