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 Izv. Akad. Nauk SSSR Ser. Mat., 1977, Volume 41, Issue 5, Pages 1125–1137 (Mi izv1883)

Discrete convolution operators on the quarter plane and their indices

R. V. Duduchava

Abstract: Let $\Gamma^2=\Gamma\times\Gamma$, where $\Gamma$ is the unit circle, and let $L_2^m(\Gamma^2)$ be the Hilbert space of vector-valued functions $\varphi=(\varphi_1,…,\varphi_m)$ whose components $\varphi_k(\zeta_1,\zeta_2)$ are complex-valued square integrable functions on $\Gamma^2$. The author considers the subspace $H_2^m(\Gamma^2)$ of functions in $L_2^m(\Gamma^2)$ having analytic continuations into the torus $\{(z_1,z_2):|z_k|<1\}$; let $P$ be the projection of $L_2^m(\Gamma^2)$ onto $H_2^m(\Gamma^2)$. For a bounded measurable matrix-valued function $a(\zeta_1,\zeta_2)$ of order $m$ on $\Gamma^2$ having limits $a(\zeta\pm0,t)$ and $a(t,\zeta\pm0)$ ($\zeta\in\Gamma)$ uniform in $t\in\Gamma$, the bounded operator $T_a^2=PaP$ is defined in $H_2^m(\Gamma^2)$. In this paper a homotopy method is described for computing the index of Noetherian operators in the $C^*$-algebra generated by the operators $T_a^2$. In the case where $a(\zeta_1,\zeta_2)$ is continuous a simple formula for computing the index of $T_a^2$ is indicated.
Bibliography: 24 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1977, 11:5, 1072–1084

Bibliographic databases:

UDC: 513.88
MSC: 47B35

Citation: R. V. Duduchava, “Discrete convolution operators on the quarter plane and their indices”, Izv. Akad. Nauk SSSR Ser. Mat., 41:5 (1977), 1125–1137; Math. USSR-Izv., 11:5 (1977), 1072–1084

Citation in format AMSBIB
\Bibitem{Dud77} \by R.~V.~Duduchava \paper Discrete convolution operators on the quarter plane and their indices \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1977 \vol 41 \issue 5 \pages 1125--1137 \mathnet{http://mi.mathnet.ru/izv1883} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=493484} \zmath{https://zbmath.org/?q=an:0406.47027|0426.47031} \transl \jour Math. USSR-Izv. \yr 1977 \vol 11 \issue 5 \pages 1072--1084 \crossref{https://doi.org/10.1070/IM1977v011n05ABEH001759} 

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This publication is cited in the following articles:
1. Albrecht Böttcher, Hartmut Wolf, “Large Sections of Bergman Space Toeplitz Operators with Piecewise Continuous Symbols”, Math Nachr, 156:1 (1992), 129
2. Albrecht Böttcher, Bernd Silbermann, “Infinite Toeplitz and Hankel Matrices with Operator-Valued Entries”, SIAM J Math Anal, 27:3 (1996), 805
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