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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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    $\begingroup$ 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 Choi
    Commented Sep 12, 2019 at 12:44
  • $\begingroup$ Unfortunately, no restriction is permitted. $\endgroup$
    – ABB
    Commented Sep 12, 2019 at 13:15
  • $\begingroup$ 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 Choi
    Commented Sep 12, 2019 at 13:22

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