Abstract
One of the goals of noncommutative geometry is to translate the basic notions of analysis into the language of Banach algebras. This translation is based on the quantization procedure. The arising operator calculus is called, following Connes, the quantum calculus. In this paper we give several assertions from this calculus concerning the interpretation of Schatten ideals of compact operators in a Hilbert space in terms of function theory. The main focus is on the case of Hilbert-Schmidt operators.
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References
L. V. Ahlfors, Conformal Invariants: Topics in Geometric Function Theory (McGraw-Hill, New York, 1973).
A. Connes, Noncommutative Geometry (Academic, San Diego, CA, 1994).
V. V. Peller, Hankel Operators and Their Applications (Springer, New York, 2003).
Acknowledgments
I am grateful to G. A. Karapetyan and A. N. Karapetyants, my colleagues from the Russian-Armenian RFBR grant 18-51-05009, with whom I discussed the idea of this paper and its results.
Funding
The work was supported in part by the Russian Foundation for Basic Research, project nos. 16-01-00117, 16-52-12012, and 18-51-05009.
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This article was submitted by the author simultaneously in Russian and English
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2019, Vol. 306, pp. 227–234.
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Sergeev, A.G. Quantum Calculus and Ideals in the Algebra of Compact Operators. Proc. Steklov Inst. Math. 306, 212–219 (2019). https://doi.org/10.1134/S0081543819050183
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DOI: https://doi.org/10.1134/S0081543819050183