C*-Algebras of Integral Operators with Piecewise Slowly Oscillating Coefficients and Shifts Acting Freely期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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C*-Algebras of Integral Operators with Piecewise Slowly Oscillating Coefficients and Shifts Acting Freely

Authors:

Institution:

(1) Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049 - 001 Lisboa, Portugal;(2) Departamento de Matemática Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2825 Monte de Caparica, Portugal;(3) Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, Col. Chamilpa, C.P. 62210 Cuernavaca, Morelos, México

Abstract:

We establish a symbol calculus for the C*-subalgebra $\mathcal{A}$ of $\mathcal{B}\left( {L^2 \left( \mathbb{T} \right)} \right)$ generated by the operators of multiplication by slowly oscillating and piecewise continuous functions and the operators $e_{h,\lambda } S_\mathbb{T} e_{h,\lambda }^{ - 1} I\left( {h \in \mathbb{R},\lambda \in \mathbb{T}} \right)$ where $S_{\mathbb{T}}$ is the Cauchy singular integral operator and $e_{{h,\lambda }} (t): = \exp {\left( {h(t + \lambda )/(t - \lambda )} \right)},t \in \mathbb{T}\backslash {\left\{ \lambda \right\}}.$ The C*-algebra $\mathcal{A}$ is invariant under the transformations

$A \mapsto U_{z} A{\user1{U}}^{{{\user1{ - 1}}}}_{{\user1{z}}} \quad {\text{and}}\quad A \mapsto e_{{h,\lambda }} Ae^{{ - 1}}_{{h,\lambda }} I{\left( {z,\lambda \in \mathbb{T},h \in \mathbb{R}} \right)},$

where U_z is the rotation operator ${\left( {U_{z} \varphi } \right)}(t): = \varphi (zt),t \in \mathbb{T}.$ Using the localtrajectory method, which is a natural generalization of the Allan-Douglas local principle to nonlocal type operators, we construct symbol calculi and establish Fredholm criteria for the C*-algebra $\mathfrak{B}$ generated by the operators $A \in \mathcal{A}$ and $U_{z} (z \in \mathbb{T}),$ for the C*-algebra $\mathfrak{C}$ generated by the operators $A \in \mathcal{A}$ and $e_{{h,\lambda }} I{\left( {h \in \mathbb{R},\lambda \in \mathbb{T}} \right)},$ and for the C*-algebra $\mathfrak{D}$ generated by the algebras $\mathfrak{B}$ and $\mathfrak{C}.$ The C*-algebra $\mathfrak{B}$ can be considered as an algebra of convolution type operators with piecewise slowly oscillating coefficients and shifts acting freely.

Keywords:

Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) Primary 47A53 47G10 47L15 Secondary 47A67 47B33 47L30

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