Sharper uncertainty principles in quaternionic Hilbert spaces. (English) Zbl 1463.47111
Summary: The uncertainty principle for quaternionic linear operators in quaternionic Hilbert spaces is established, which generalizes the result of Goh-Micchelli. It turns out that there appears an additional term given by a commutator that reflects the feature of quaternions. The result is further strengthened when one operator is self-adjoint, which extends under weaker conditions the uncertainty principle of Dang-Deng-Qian from complex numbers to quaternions. In particular, our results are applied to concrete settings related to quaternionic Fock spaces, quaternionic periodic functions, quaternion Fourier transforms, quaternion linear canonical transforms, and nonharmonic quaternion Fourier transforms.
MSC:
| 47B47 | Commutators, derivations, elementary operators, etc. |
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 47A05 | General (adjoints, conjugates, products, inverses, domains, ranges, etc.) |
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