Abstract
For a bounded pseudo-differential operator with the dense domain \(C^\infty(\mathbb{S}^1)\) on \(L^p(\mathbb{S}^1)\) , the minimal and maximal operator are introduced. An analogue of Agmon-Douglis-Nirenberg [1] is proved and then is used to prove the uniqueness of the closed extension of an elliptic pseudo-differential operator of symbol of positive order. We show the Fredholmness of the minimal operator. The essential spectra of pseudo-differential operators on \(\mathbb{S}^1\)are described.
Mathematics Subject Classification (2000). Primary 47G30
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References
S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math. 12 (1959), 623–727.
M.S. Agranovich, Spectral properties of elliptic pseudodifferential operators on a closed curve, Funktsional. Anal. i Prilozhen. 13 (1979), 54–56 (in Russian).
M.S. Agranovich, Elliptic pseudodifferential operators on a closed curve, Trudy Moskov. Mat. Obshch. 47 (1984), 22–67, 246 (in Russian); Trans. Moscow Math. Soc. (1985), 23–74.
B.A. Amosov, On the theory of pseudodifferential operators on the circle, Uspekhi Mat. Nauk. 43 (1988), 169–170 (in Russian); Russian Math. Surveys 43 (1988), 197– 198.
A. Dasgupta and M.W. Wong, Spectral theory of SG pseudo-differential operators on Lp(Rn), Studia Math. 187 (2008), 185–197.
S. Molahajloo and M.W. Wong, Pseudo-differential operators on S1, in New Developments in Pseudo-Differential Operators, Operator Theory: Advances and Applications 189, Birkh¨auser, 2008, 297–306.
S. Molahajloo and M.W. Wong, Ellipticity, Fredholmness and spectral invariance of pseudo-differential operators on S1, J. Pseudo-Differ. Oper. Appl. 1 (2010), 183–205.
F. Nicola and L. Rodino, SG pseudo-differential operators and weak hyperbolicity, Pliska Stud. Math. Bulgar. 15 (2002), 5–19.
M. Ruzhansky and V. Turunen, On the Fourier analysis of operators on the torus, in Modern Trends in Pseudo-Differential Operators, Operator Theory: Advances and Applications 172, Birkh¨auser, 2007, 87–105.
M. Ruzhansky and V. Turunen, Pseudo-Differential Operators and Symmetries, Birkh¨auser, 2009.
M. Schechter, On the essential spectrum of an arbitrary operator I, J. Math. Anal. Appl. 13 (1966), 205–215.
M. Schechter, Spectra of Partial Differential Operators, Second Edition, North- Holland, 1986.
E. Schrohe, Spectral invariance, ellipticity, and the Fredholm property for pseudodifferential operators on weighted Sobolev spaces, Ann. Global Anal. Geom. 10 (1992), 237–254.
F. Wolf, On essential spectrum of partial differential boundary problems, Comm. Pure Appl. Math. 12 (1959), 211–228.
M.W. Wong, An Introduction to Pseudo-Differential Operators, Second Edition, World Scientific, 1999.
M.W. Wong, Fredholm pseudo-differential operators on weighted Sobolev spaces, Ark. Mat. 21 (1983), 271–282.
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Pirhayati, M. (2011). Spectral Theory of Pseudo-Differential Operators on \(\mathbb{S}^1\) . In: Rodino, L., Wong, M., Zhu, H. (eds) Pseudo-Differential Operators: Analysis, Applications and Computations. Operator Theory: Advances and Applications(), vol 213. Springer, Basel. https://doi.org/10.1007/978-3-0348-0049-5_2
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DOI: https://doi.org/10.1007/978-3-0348-0049-5_2
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