# Unboundedness of pseudo-inverse

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?

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