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

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

2005-01-10

arxiv.org/abs/math/0501139v7

Mathematics

Functional Analysis

8 pages, minor revision, to appear in Appl. Anal. Disc. Math

Scientific paper

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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