Communications in Mathematical Analysis

Algebras of Pseudodifferential Operators with Discontinuous Oscillating Symbols.

Yu. I. Karlovich, V. S. Rabinovich, and N. L. Vasilevski

Full-text: Open access

Abstract

Non-closed algebras ${\mathfrak{A}}_{\delta,\gamma}$ of pseudodifferential operators with slowly oscillating Lipschitz symbols in $\Lambda^{SO}_{\delta,\gamma}({\mathbb R}\times{\mathbb R})$ with $\delta,\gamma\in(1/2,1]$ and the minimal $C^*$-algebra ${\mathfrak{A}}$ containing all ${\mathfrak{A}}_{\delta,\gamma}$ are studied on the Lebesgue space $L^2({\mathbb R})$. Applying results on the boundedness and compactness of pseudodifferential operators $A\in{\mathfrak{A}}$, a commutative algebra of their Fredholm symbols is described. A Fredholm criterion and an index formula for the operators $A\in{\mathfrak{A}}$ are obtained. Then we study the Fredholmness for the $C^*$-algebra ${\mathfrak{B}}$ generated by the operators $A\in{\mathfrak{A}}$ with multiplicatively slowly oscillating Lipschitz symbols and by the operators of multiplication $aI$ and convolution operators $W^0(b)$ with piecewise continuous functions $a,b:\overline{{\mathbb R}}\to{\mathbb C}$. The algebra of Fredholm symbols for the operators $A\in{\mathfrak{B}}$ is not commutative. A Fredholm criterion for the operators $A\in{\mathfrak{B}}$ is established.

Article information

Source
Commun. Math. Anal., Conference 3 (2011), 108-130.

Dates
First available in Project Euclid: 25 February 2011

Permanent link to this document
https://projecteuclid.org/euclid.cma/1298670007

Mathematical Reviews number (MathSciNet)
MR2772056

Zentralblatt MATH identifier
1218.47135

Subjects
Primary: 47G30: Pseudodifferential operators [See also 35Sxx, 58Jxx]
Secondary: 47A53: (Semi-) Fredholm operators; index theories [See also 58B15, 58J20] 47G10: Integral operators [See also 45P05]

Keywords
Pseudodifferential operator $C^*$-algebra slowly oscillating Lipschitz symbol discontinuous oscillating symbol oscillatory integral boundedness compactness

Citation

Karlovich, Yu. I.; Rabinovich, V. S.; Vasilevski, N. L. Algebras of Pseudodifferential Operators with Discontinuous Oscillating Symbols. Commun. Math. Anal. (2011), no. 3, 108--130. https://projecteuclid.org/euclid.cma/1298670007


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