Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space $F$.

Assume that $S$ is invertible. Is it true that $S$ is normal iff $S^{-1}$ is normal?

If $S$ is normal then $SS^*=S^*S$ and thus $(SS^*)^{-1}=(S^*S)^{-1}$. Hence $(S^{-1})^*S^{-1}=S^{-1}(S^{-1})^*$

  • $\begingroup$ Yes, it is true, and yes the statements below prove the claim (or at least half of it). $\endgroup$ – Aweygan Apr 4 '18 at 14:24
  • $\begingroup$ @Aweygan Why the converse is true? Thank you $\endgroup$ – Schüler Apr 4 '18 at 14:32

If $S^{-1}$ is normal, then $$(S^*S)^{-1}=S^{-1}(S^{-1})^*=(S^{-1})^*S^{-1}=(SS^*)^{-1}.$$ Thus $S^*S=SS^*$ and therefore $S$ is normal.

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