Abstract
We give necessary and sufficient conditions on the symbols to guarantee that the corresponding pseudo-differential operators are nuclear from \(L^{p_1}({\mathbb {S}}^1)\) into \(L^{p_2}({\mathbb {S}}^1)\) for \(1\le p_1,p_2<\infty \). Applications are given to adjoints of nuclear pseudo-differential operators from \(L^{p_2'}({\mathbb {S}}^1)\) into \(L^{p_1'}({\mathbb {S}}^1)\) for \(1\le p_1,p_2<\infty \) and products of nuclear pseudo-differential operators on \(L^p({\mathbb {S}}^1),\,1\le p<\infty \).
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References
Cardona, D.: Hölder estimates for pseudo-differential operators on \({\mathbb{T}}^1\). J. Pseudo-Differ. Oper. Appl. 5, 517–525 (2014)
Delgado, J.: The trace of nuclear operators on \(L^p(\mu )\) for \(\sigma \)-finite Borel measureson second countable spaces. Integral Equ. Oper. Theory 68, 61–74 (2010)
Delgado, J.: A trace formula for nuclear operators on \(L^p\), In: Schulze, B.W., Wong, M.W. (eds.) Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations, Operator Theory: Advances and Applications, vol. 205, pp. 181–193. Birkhäuser, Basel (2010)
Delgado, J., Ruzhansky, M.: Kernel and symbol criteria for Schatten classes and \(r\)-nuclarity on compact manifolds. C. R. Acad. Sci. Paris Ser. I 352, 779–784 (2014)
Delgado, J., Ruzhansky, M.: \(L_p\) nuclarity, traces, and Grothendieck-Lidskii formula on compact Lie groups. J. Math. Pures Appl. 102, 153–172 (2014)
Delgado, J., Wong, M.W.: \(L^p\)-nuclear pseudo-differential operators on \(\mathbb{Z}\) and \(\mathbb{S}^1\). Proc. Am. Math. Soc. 141, 3935–3942 (2013)
Ghaemi, M.B., Nabizadeh Morsalfard, E., Jamalpour Birgani, M.: A study on the adjoint of pseudo-differential operators on \(\mathbb{S}^1\) and \(\mathbb{Z}\). J. Pseudo-Differ. Oper. Appl. 6, 197–203 (2015)
Ghaemi, M.B., Jamalpour Birgani, M., Nabizadeh Morsalfard, E.: A study on pseudo-differential operators on \(\mathbb{S}^1\) and \(\mathbb{Z}\). J. Pseudo-Differ. Oper. Appl. 7, 237–247 (2016)
Gohberg, I., Goldberg, S., Krupnik, N.: Traces and Determinants of Linear Operators, Operator Theory: Advances and Applications, vol. 116. Birkhäuser, Basel (2000)
Grothendieck, A.: Produits Tensoriels Topologiques et Espaces Nucléaires. Memoirs Amer. Math. Soc. 16, (1955)
Grothendieck, A.: La theorie de Fredholm. Bull. Soc. Math. Fr. 84, 319–384 (1956)
Molahajloo, S.: A characterization of compact pseudo-differential operators on \({\mathbb{S}}^1\), In: Rodino, L., Wong, M.W., Zhu, H. (eds.) Pseudo-Differential Operators: Analysis, Applications and Computations, Operator Theory: Advances and Applications, vol. 213, pp. 25–31. Birkhäuser, Basel (2011)
Molahajloo, S., Wong, M. W.: Pseudo-differential operators on \(\mathbb{S}^1\), In: Rodino, L., Wong, M.W. (eds.) New Developments in Pseudo-Differential Operators, Operator Theory: Advances and Applications, vol. 189, pp. 297–306. Birkhäuser, Basel (2008)
Molahajloo, S., Wong, M.W.: Ellipticity, Fredholmness and spectral invariance of pseudo-differential operators on \(\mathbb{S}^1\). J. Pseudo-Differ. Oper. Appl. 1, 183–205 (2010)
Pirhayati, M.: Spectral theory of pseudo-differential operators on \(\mathbb{S}^1\), In: Rodino, L., Wong, M.W., Zhu, H. (eds.) Pseudo-Differential Operators: Analysis, Applications and Computations, Operator Theory: Advances and Applications, vol. 213, pp. 15–25. Birkhäuser, Basel (2011)
Wong, M. W.: Trace class Weyl transforms, In: Recent Advances in Operator Theory and its applications, Operator Theory: Advances and Applications, vol. 160, pp. 469–478. Birkhäuser, Basel (2005)
Wong, M.W.: Discrete Fourier Analysis. Birkhäuser, Basel (2011)
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The authors are grateful to the referee for the comments on the first version of the paper.
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The research of M. Jamalpour Birgani was carried out and completed during his visit of Professor M. W. Wong under the auspices of the International Visiting Research Traineeship (IVRT) in the Department of Mathematics and Statistics at York University.
M. W. Wong: This research has been partially supported by the Natural Sciences and Engineering Research Council of Canada under Discovery Grant 0008562.
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Ghaemi, M.B., Birgani, M.J. & Wong, M.W. Characterizations of nuclear pseudo-differential operators on \({\mathbb {S}}^1\) with applications to adjoints and products. J. Pseudo-Differ. Oper. Appl. 8, 191–201 (2017). https://doi.org/10.1007/s11868-017-0199-7
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DOI: https://doi.org/10.1007/s11868-017-0199-7