Let $X$ be a Hilbert space. Let $T : X \rightarrow X$ be a linear mapping.
Suppose we have two scalar products $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ on $X$.
Let $T_1$ and $T_2$ the pseudoinverses of $T$ with respect to these two scalar products, respectively.
Can we estimate $\| T_1 \|$ in terms of $\| T_2 \|$?