Traces of pseudo-differential operators on compact and Hausdorff groups

We give a characterization of and a trace formula for trace class pseudo-differential operators on compact Hausdorff groups.

The aim of this paper is to give a characterization of trace class pseudo-differential operators on compact and Hausdorff groups. We give a formula for the trace of pseudodifferential operators in the trace class. The main technique is to obtain a formula for the symbol of the product of two pseudo-differential operators on a compact and Hausdorff group.
In Sect. 2, we give a brief recall of Hilbert-Schmidt and trace class operators. We define pseudo-differential operators on compact and Hausdorff groups by using the space of all unitary and irreducible representations in Sect. 3. A product formula for pseudo-differential operators is given. Then we give a characterization of trace class pseudo-differential operators.

Hilbert-Schmidt and trace class operators
Let H be a complex and separable Hilbert space in which the inner product and norm are denoted by (, ) H and H . Let A : H → H be a compact operator. Then the absolute value of A denoted by |A| is defined by The operator |A| is compact and positive. So, by the spectral theorem, there exists an orthonormal basis {ϕ j } ∞ j=1 of eigenvectors of |A| with the corresponding eigenvalues {s j } ∞ j=1 of real numbers. The operator A is said to be in the Hilbert-Schmidt class and we define its Hilbert-Schmidt norm by The operator A is said to be in the trace class and we define its trace by We have the following theorem which shows that the Hilbert-Schmidt norm and trace is independent of the choice of orthonormal basis for H, for details see [12] by Reed and Simon.
The following theorem will be useful in the next section.
A unitary representation π : G → U (H) is called irreducible if it has only the trivial invariant subspaces, i.e., {0} and H. The following theorem is crucial in defining pseudo-differential operators on compact Hausdorff groups. For its proof see [6,20].

Theorem 3.1 Let G be a compact and Hausdorff group. Then any irreducible and unitary representation of G on a complex and separable Hilbert space is finite dimensional.
Let G be a compact and Hausdorff group with the left (and right) Haar measure is denoted by μ. LetĜ be the set of all irreducible and unitary representations of G.
Let π ∈Ĝ. Then π is finite dimensional. Let a π be the degree of π and let H π be its representation space. We denote the inner product and the norm in H π by ( , ) H π and H π respectively. Let {ψ 1 , ψ 2 , . . . , ψ a π } be an orthonormal basis for H π . Then for all j, k = 1, 2, . . . , we define π jk by . . , a π } forms an orthonormal basis for L 2 (G).
By Peter-Weyl theorem, every f ∈ L 2 (G) can be expressed as have the following Plancheral theorem.
T σ is called the pseudo-differential operator on G corresponding to the symbol σ . For pseudo-differential operators on Euclidean spaces see for example [15,19].
Let L 2 (G×Ĝ×N×N) be the space all measurable functions σ on G ×Ĝ × N × N for which We have the following result on the L 2 -boundedness of pseudo-differential operators on G.
where * is the norm in the C * -algebra of all bounded linear operators from L 2 (G) into L 2 (G).
Proof Let f ∈ L 2 (G). By Minkowski's inequality and the Schwarz inequality, we have Now we are ready to give a necessary and sufficient condition on σ to guarantee Hilbert-Schmidt properties of T σ .
Hence π ∈Ĝ a π j,k=1 . and the proof is complete.
Let σ and τ measurable functions on G ×Ĝ × N × N. Then we define σ τ by for all g ∈ G, ξ ∈Ĝ and l, m = 1, . . . , a ξ . The following theorem shows that that the composition of two pseudo-differential operators on G is again a pseudo-differential operator.
We can now look at the special case when G = S 1 .
Then S 1 = {π n : n ∈ Z}, and S Z. Hence for any symbol σ on S 1 × Z we can define pseudo-differential operator T σ on L 2 (S 1 ) by (T σ f ) (θ ) = n∈Z σ (θ, n)( f, π n ) L 2 (S 1 ) π n (θ ), f ∈ L 2 (S 1 ), θ ∈ [−π, π]. Hence, Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.