On the Algebra Generated by Volterra Integral Operators with Homogeneous Kernels and Continuous Coefficients

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/p3569-9057-4562-o On the Algebra Generated by Volterra Integral Operators with Homogeneous Kernels and Continuous Coefficients Avsyankin, O. G. , Kamenskikh, G. A. Vladikavkaz Mathematical Journal 2022. Vol. 24. Issue 4. Abstract: We consider Volterra multidimensional integral operators with continuous coefficients in Lebesgue spaces. It is assumed that the kernel of the integral operator is homogeneous of degree \((-n)\), invariant under the rotation group \(SO(n)\) and satisfies a certain summability condition that ensures the boundedness of the operator. In this paper, the main object of research is the Banach algebra \(\mathfrak{A}\) generated by all operators of the above type and the identity operator. The algebra \(\mathfrak{A}\) is noncommutative, and for its study we turn to the quotient algebra \(\mathfrak{A}/\mathfrak{T}\), where \(\mathfrak{T}\) is the set of all compact operators. It is shown that the algebra \(\mathfrak{A}/\mathfrak{T}\) is commutative, which makes it possible to apply the general methods for studying commutative Banach algebras. In particular, a description of the maximal ideals space of the algebra \(\mathfrak{A}/\mathfrak{T}\) is given and a criterion for the invertibility of elements from this algebra is found. Based on this, we construct a symbolic calculus for the Banach algebra \(\mathfrak{A}\) that is, each operator from this algebra is assigned a certain continuous function. This function is called the symbol of the operator. In terms of the symbol, we obtained necessary and sufficient conditions for the Fredholm property of an operator from \(\mathfrak{A}\), as well as an index formula. Keywords: integral operator, homogeneous kernel, symbol, Fredholmness, index, Banach algebra Language: Russian Download the full text For citation: Avsyankin, O. G. and Kamenskikh, G. A. On the Algebra Generated by Volterra Integral Operators with Homogeneous Kernels and Continuous Coefficients, Vladikavkaz Math. J., 2022, vol. 24, no. 4, pp.19-29 (in Russian). DOI 10.46698/p3569-9057-4562-o + References 1. Karapetiants, N. and Samko, S. Equations with Involutive Operators, Boston, Basel, Berlin, Birkhauser, 2001, 427 p. 2. Avsyankin, O. G. and Karapetyants, N. K. On the Pseudospectra of Multidimensional Integral Operators with Homogeneous Kernels of Degree \(-n\), Siberian Mathematical Journal, 2003, vol. 44, no. 6, pp. 935-950. DOI: 10.1023/B:SIMJ.0000007469.86630.6b. 3. Avsyankin, O. G. On the \(C^\)-Algebra Generated by Multidimensional Integral Operators with Homogeneous Kernels and Multiplicative Translations, Doklady Mathematics,* 2008, vol. 77, no. 2, pp. 298-299. DOI: 10.1134/S106456240802035X. 4. Avsyankin, O. G. Volterra Type Integral Operators with Homogeneous Kernels in Weighted \(L_p\)-Spaces, Russian Mathematics, 2017, vol. 61, no. 11, pp. 1-9. DOI: 10.3103/S1066369X17110019. 5. Avsyankin, O. G. Invertibility of Multidimensional Integral Operators with Bihomogeneous Kernels, Mathematical Notes, 2020, vol. 108, no. 2, pp. 277-281. DOI: 10.1134/S0001434620070287. 6. Avsyankin, O. G. On Integral Operators with Homogeneous Kernels and Trigonometric Coefficients, Russian Mathematics, 2021, vol. 65, no. 4, pp. 1-7. DOI: 10.3103/S1066369X21040010. 7. Umarkhadzhiev, S. M. One-Sided Integral Operators with Homogeneous Kernels in Grand Lebesgue Spaces, Vladikavkaz Mathematical Journal, 2017, vol. 19, no. 3, pp. 70-82 (in Russian). 8. Mikhailov, L. G. New Class of Singular Integral Equations and Its Application to Differential Equations with Singular Coefficients, Dushanbe, Tr. Akad. Nauk Tadzhik. SSR, 1963, vol. 1, 183 p. (in Russian). 9. Avsyankin, O. G. Development of the Theory of Multidimensional Integral Operators with Homogeneous and Bihomogeneous Kernels: Dr. Science Thesis, Rostov-on-Don, South Federal University, 2009, 277 p. (in Russian). 10. Gelfand, I. M., Raikov, D. A. and Shilov, G. E. Commutative Normed Rings, Moscow, Fizmatgiz, 1960, 315 p. (in Russian). The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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