Let $H$ be a Hilbert space and consider bounded operators $a$ and $b$ on $H$.
For given operators $a$ and $b$, I am looking a way to get all solutions (bdd operators $x$) of the inequality $\|xb-a\|<1$ in termes of $a$ and $b$.
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1$\begingroup$ This question would seem more sensible if you had certain restrictions on the norm of $a$ and the norm of $b$. Otherwise, if $\Vert a\Vert <1$ then I always get trivial solutions for any $b$, by taking $\Vert x\Vert$ to be very very small $\endgroup$– Yemon ChoiCommented Sep 12, 2019 at 12:44
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$\begingroup$ Unfortunately, no restriction is permitted. $\endgroup$– ABBCommented Sep 12, 2019 at 13:15
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$\begingroup$ Well, then I don't think you can do anything, unless you start to consider possible cases. As I said above, if $a$ has small norm then any $x$ which has sufficiently small norm will work, and none of the intuition from the "solve $xb=a$" problem will be valid $\endgroup$– Yemon ChoiCommented Sep 12, 2019 at 13:22
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