Traces of pseudo-differential operators on $$\mathbb{S }^{n-1}$$
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Traces of pseudo-differential operators on Sn−1 Zhi Chen · M. W. Wong
Received: 15 November 2012 / Revised: 29 December 2012 / Accepted: 1 January 2013 / Published online: 10 January 2013 © Springer Basel 2013
Abstract We give a characterization of and a trace formula for trace class pseudodifferential operators on the unit sphere Sn−1 centered at the origin in Rn . Keywords Pseudo-differential operators on Sn−1 · Hilbert–Schmidt operators · Trace class operators · Traces · Spherical harmonics Mathematics Subject Classification (2010)
Primary 47G30; Secondary 42A45
1 Introduction In the paper [5] and the book [9], a characterization of Hilbert–Schmidt pseudodifferential operators on the unit circle S1 centered at the origin is given. The condition given on the symbol is L 2 in nature and is different from Hörmander’s S m condition that is prevalent in the study of partial differential equations [4,8]. The L 2 and the related L p , 1 ≤ p ≤ ∞, conditions on the symbols allow singularities and are ideal for a broad spectrum of disciplines ranging from functional analysis to operator algebras and to time-frequency analysis. Trace class pseudo-differential operators on L p (S1 ) and their traces can be found in the paper [3].
This research has been supported by the Natural Sciences and Engineering Research Council of Canada. Z. Chen · M. W. Wong (B) Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canada e-mail: [email protected] Z. Chen e-mail: [email protected]
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Z. Chen, M. W. Wong
The aim of this paper is to give a characterization of trace class pseudo-differential operators on the unit sphere Sn−1 with center at the origin in Rn and we give a formula for the trace of each such trace class operator. Spherical harmonics on Sn−1 are the natural analogs of Fourier series when we pass from the circle S1 to the sphere Sn−1 . The main technique is to obtain a formula for the symbol of the product of two pseudodifferential operators Tσ and Tτ on Sn−1 . We give a brief recall of Hilbert–Schmidt operators and trace class operators in Sect. 2. In Sect. 3 is a summary of results on spherical harmonics on Sn−1 that we need in this paper. We give a product formula for pseudo-differential operators, a characterization of trace class pseudo-differential operators and then a trace formula for pseudo-differential operators on S1 in Sect. 4. The same is done for Sn−1 in Sect. 5. The study of trace class operators and their traces can best be put in the context of noncommutative measure theory in which trace class operators are the noncommutative analogs of measurable functions and traces are noncommutative analogs of integrals. See the book [2] by Alain Connes in this connection. 2 Hilbert–Schmidt operators and trace class operators Let X be a complex and separable Hilbert space in which the inner product and norm are denoted by ( , ) X and X respectively. Let A : X → X be a compact operator. Then the absolute value |A| of the operator A is defined by |A| = (A
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