The Dynamical Systems (A, G, ω) and Pseudo-differential Operators on Locally Compact Commutative Groups

Author:

Abstract: In the first part, we consider dynamical systems $$(A, G, ω)$$ locally compact Abelian group, we generalize the Takesaki-Takai theorem and show that the τ-double-crossed product $$(Ax^{τ}_{ω} G)x^{τ}_{ω}\hat{G}$$ is isomorphic to the tensor product $$A\otimes LK (L^{2}(G))$$, where the space $$LK (L^{2}(G))$$ of all compact operators on $$L^{2}(G)$$. In the second part, we introduce the $$\Im$$-Wigner function, which generates pseudo-differential operators $$Op_{\Im}(α)$$ by association $$\langle Op_{\Im}(α) ψ,φ \rangle_{L^{2}(G)} = \langle \hat{a}, W_{\Im}(ψ,φ) \rangle _{B^{2}(\hat{G}xG)}$$. We establish that assume $$Op_{\Im}(α)$$ is a bounded pseudo-differential operator $$L^{2}(G)\otimes B^{2}(\hat{G})\rightarrow B(L^{2}(G))$$ then operator $$Op_{\Im}(α)$$ extends to topological isomorphism $$\wp ' (G)\otimes (F ( \wp (\hat{G})))^{'} \rightarrow B(\wp(G), \wp ' (G))$$, where $$\wp (G)$$ is the Bruhat space $$\wp (G)$$.

Equations, E-ISSN: 2944-9146, Volume 3, 2023, Art. #8

PDF
DOI
XML
Cite
×
Citation Tools
WSEAS AMA
Mykola Yaremenko, "The Dynamical Systems (A, G, ω) and Pseudo-differential Operators on Locally Compact Commutative Groups," Equations, vol. 3, pp. 59-67, 2023, DOI:10.37394/232021.2023.3.8
Mykola Yaremenko. The Dynamical Systems (A, G, ω) and Pseudo-differential Operators on Locally Compact Commutative Groups. Equations. 2023;3:59-67. 10.37394/232021.2023.3.8
Copy to Clipboard
Citation copied to Clipboard

Certification

---