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Superstability of adjointable mappings on Hilbert $C^*$-modules

Michael Frank·Pasc Gavruta·Mohammad Sal Moslehian
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Abstract

We define the notion of $\varphi$-perturbation of a densely defined adjointable mapping and prove that any such mapping $f$ between Hilbert ${\mathcal A}$-modules over a fixed $C^*$-algebra ${\mathcal A}$ with densely defined corresponding mapping $g$ is ${\mathcal A}$-linear and adjointable in the classical sense with adjoint $g$. If both $f$ and $g$ are everywhere defined then they are bounded. Our work concerns with the concept of Hyers--Ulam--Rassias stability originated from the Th.M. Rassias' stability theorem that appeared in his paper [On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297--300]. We also indicate interesting complementary results in the case where the Hilbert $C^*$-modules admit non-adjointable $C^*$-linear mappings.

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Michael Frank, Pasc Gavruta, Mohammad Sal Moslehian. 2005-01-10. Superstability of adjointable mappings on Hilbert $C^*$-modules. https://doi.org/10.2298/aadm0901039f

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