Norm estimate for Moore-Penrose pseudo-inverse of $i^\ast T i$

2012-07-31T08:15:31Z

Let $G$ and $H$ be Hilbert spaces, let $i : G \rightarrow H$ be an isometric inclusion (so $G$ is a subspace of $H$) and let $T : H \rightarrow H$ be a bounded linear operator with closed range.

That $T$ has closed range $R(T)$ is equivalent to the existence of a constant $C > 0$ such that

$\forall y \in R(T) : \exists x \in T^{-1}({y}) : C \| y \|_H \geq \|x\|$

Another equivalent condition is that the Moore-Penrose inverse of $T$, written $T^\dagger$, has a norm bounded by the constant $C$.

Do you know a norm bound for $(i^\ast T i)^\dagger$? You assume that $G$ is finite dimensional so that $i^\ast T i$ has closed range.

Norm estimate for Moore-Penrose pseudo-inverse of $i^\ast T i$ - MathOverflow

Norm estimate for Moore-Penrose pseudo-inverse of $i^\ast T i$