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Let $H$ be a real Hilbert space, and let $T$ be a bounded self-adjoint operator of $H$ to itself. Let $T^+$ denote the pseudo-inverse operator of $T$, that is the operator defined to be zero on $(TH)^{\perp}$ and the inverse of $T$ on $TH$.

Suppose now that for all increasing sequences $H_1\subset H_2\subset \dots$ of finite dimensional subspaces of $H$ we have $\|(T|_{H_i})^+\|\to \infty$ as $i\to \infty$. Is this enough to conclude that $T^+$ is unbounded?

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  • $\begingroup$ Why the downvote? Please leave a comment if you think the question is not formulated clearly, I am not a functional analyst... $\endgroup$ – Maurizio Moreschi Jan 24 at 7:47

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