3rd International Conference on

Applied Physics and Mathematics

Scientific Program

Mohammad Bagher Ghaemi

Mohammad Bagher Ghaemi

Department of Mathematics, Iran University of Science and Technology, Iran

Title: Compactness and nuclearity of psedifferential operators on lp(s1) and Lp(z), p greater than or equal to 1

Biography:

Mohammad Bagher Ghaemi belongs to Department of Mathematics, Iran University of Science and Technology, Iran. He completed his undergraduate course in Ferdowsi University of Mashhad from October 1983 to July 1987, graduating B.Sc. in Mathematics and Mathematical Education (first class). He completed his Master’s Course in Ferdowsi University of Mashhad from October 1987 to July 1990, graduating Ms.C. in Mathematics (first class). He did his Ms.C. thesis on "Automatic Continuity of Generalized Intertwining Operators".  He pursued his Ph.D. Course in Glasgow University (UK) from March from March 1996 to October 2000, graduating Ph.D. in functional Analysis. He did his Ph.D. Thesis on "Spectral Theory of Linear Operators".

Abstract

Compactness and nuclearity of psedifferential operators on lp(s1) and Lp(z), p ≥ 1
 
Let Z be the set of all integers and let S1  be the unit circle centered at the origin. Suppose σ be a measurable function on Z × S1. 
 
Then for every sequence in Lp(Z), 1 ≤ p < ∞, we define the sequence Tσa by
 
(Tσa)(n) = 1/2π e−inθ σ(n, θ)(FZa)(θ)dθ, n ∈ Z
 
where FZa is the Fourier transform of a given by ∞
 
(FZa)(θ) = n=∑−∞ a(n)einθ, θ ∈ [−π, π].
 
Assume τ be a measurable function on S1 × Z. Then for all f in Lp(S1), 1 ≤ p < ∞, we define the function Tτ f on S1 by
 
(Tτ f )(θ) = einθτ n∈Z (θ, n)(FS1 f )(n), θ ∈ [−π, π],
 
Functions σ and τ are called symbols of pseudo-differential operators Tσ and
Tτ on Z and S1 respectively.
 
In this article we show that if σ(θ, n) ∈ S1 × Z and there exist C > 0, m ∈ R such that for all n ∈ Z, |σ(n, θ)| ≤  C(1 + n2)mand and for all f ∈  Lp(S1)  the series n= σ(θ, n)f (n), θ ∈ [−π, π] is absolutely convergent, then the pseudo-differential operator Tσ : Lp(S1) −→  Lp(S1),  1 ≤ p < ∞ is compact and bounded. Then we prove similar results for pseudo differential operators on Z. A necessary and sufficient condition for pseudo-differential operators from Lp1 (S1) into Lp2 (S1) and from Lp1 (Z) into Lp2 (Z) to be nuclear are presented for 1 ≤ p1, p2 < ∞. In the cases when p1 = p2, the trace formulas of pseudo- differential operators based on their symbols are given.

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