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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 $\| T_2 \|$?

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I want to say no, simply because you state only that the Hilbert spaces are isomorphic. Nothing about how $S$ and $T$ are related. They would not have any other relationship even if they were in the same space. –  adam W Mar 7 '13 at 19:01
    
Normed spaces of the same finite dimension are automatically isomorphic. // There must be some relation between $S$ and $T$, otherwise nothing can be done. The question title made me think that the linear map stays the same while the norm changes. But in the actual question no relation between $S$ and $T$ is given. –  user53153 Mar 7 '13 at 19:05
    
I have completely rewritten the question. –  shuhalo Mar 7 '13 at 19:34
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