Traces of pseudo-differential operators on compact and Hausdorff groups

Abstract

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

1 Introduction

Pseudo-differential operators, first developed by Kohn and Nirenberg [8] in 1965 and then is used by Hörmander [7] and others for problems in partial differential equations. Weyl transforms which are a class of pseudo-differential operators have applications in Quantization due to Hermann Weyl [18]. In [21] Weyl transforms on compact Lie groups are introduced and the heat kernels of the Laplacian on the compact Lie group is obtained. In this paper, we look at pseudo-differential operators on compact and Hausdorff groups with $L^2$ symbols. The $L^p,$ $1\le p\le \infty$ conditions on the symbols allowing singularities are ideal for a broad spectrum of disciplines ranging from functional analysis to operator algebras to quantization. An analogue of the results in this paper is studied for the compact Lie group $\mathbb S ^{n-1}$, i.e., the unit sphere with center at the origin in [5]. Pseudo-differential operators on the unit sphere are studied extensively in [1–11, 13, 14, 16, 17, 22].

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 pseudo-differential 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.

2 Hilbert-Schmidt and trace class operators

Let $\mathcal{H}$ be a complex and separable Hilbert space in which the inner product and norm are denoted by $(,)_{\mathcal{H}}$ and $\Vert \ \Vert _{\mathcal{H}}$. Let $A:\mathcal{H}\rightarrow \mathcal{H}$ be a compact operator. Then the absolute value of A denoted by $|A|$ is defined by

$$\begin{aligned} |A|=(A^*A)^{1/2}. \end{aligned}$$

The operator $|A|$ is compact and positive. So, by the spectral theorem, there exists an orthonormal basis $\{\varphi _j\}_{j=1}^{\infty }$ of eigenvectors of $|A|$ with the corresponding eigenvalues $\{s_j\}_{j=1}^\infty$ of real numbers. The operator $A$ is said to be in the Hilbert-Schmidt class $S_2$ if

$$\begin{aligned} \sum _{j=1}^{\infty }s_j^2<\infty , \end{aligned}$$

and we define its Hilbert-Schmidt norm by

$$\begin{aligned} \Vert A\Vert _{HS}=\left( \sum _{j=1}^{\infty }s_j^2\right) ^{1/2}. \end{aligned}$$

The operator $A$ is said to be in the trace class $S_1$ if

$$\begin{aligned} \sum _{j=1}^\infty s_j<\infty , \end{aligned}$$

and we define its trace by

$$\begin{aligned} tr(A)=\sum _{j=1}^\infty s_j. \end{aligned}$$

We have the following theorem which shows that the Hilbert-Schmidt norm and trace is independent of the choice of orthonormal basis for $\mathcal{H}$, for details see [12] by Reed and Simon.

Theorem 2.1

Let $A:\mathcal{H}\rightarrow \mathcal{H}$ be a bounded operator. Then

• $A\in S_2$ if and only if there exists an orthonormal basis $\{\varphi _j\}_{j=1}^\infty$ for $\mathcal{H}$ such that

  $$\begin{aligned} \Vert A\Vert _{HS}=\left( \sum _{j=1}^{\infty }\Vert A\varphi _j\Vert _{\mathcal{H}}^2\right) ^{1/2}<\infty . \end{aligned}$$

• $A\in S_1$ if and only if there exists an orthonormal basis $\{\varphi _j\}_{j=1}^\infty$ for $\mathcal{H}$ such that

  $$\begin{aligned} tr(A)=\sum _{j=1}^{\infty }(A\varphi _j,\varphi _j)_{\mathcal{H}}<\infty . \end{aligned}$$

The following theorem will be useful in the next section.

Theorem 2.2

Let $A:\mathcal{H}\rightarrow \mathcal{H}$ be a bounded operator. Then $A\in S_1$ if and only if $A=BC$, where $B$ and $C$ are in $S_2$.

3 Pseudo-differential operators on compact and Hausdorff groups

Let $G$ be a locally compact and Hausdorff group. Let $\mathcal{H}$ be a separable and complex Hilbert space. We denote the group of all unitary operators by $U(\mathcal{H})$. A group homomorphism $\pi :G\rightarrow U(\mathcal{H})$ is said to be a unitary representation of $G$ on $\mathcal{H}$ if it is strongly continuous, i.e., the mapping

$$\begin{aligned} g\mapsto \pi (g)h \end{aligned}$$

is a continuous mapping for all $h\in \mathcal{H}$. The dimension of $\mathcal{H}$ is knows as the degree or the dimension of the representation $\pi$.

A closed subspace M of $\mathcal{H}$ is said to be invariant with respect to the unitary representation $\pi :G\rightarrow U(\mathcal{H})$ if

$$\begin{aligned} \pi (g)M\subseteq M,\quad \forall g\in G. \end{aligned}$$

A unitary representation $\pi :G\rightarrow U(\mathcal{H})$ is called irreducible if it has only the trivial invariant subspaces, i.e., $\{0\}$ and $\mathcal{H}.$ 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 $\mathbb{G }$ be a compact and Hausdorff group with the left (and right) Haar measure is denoted by $\mu .$ Let $\hat{\mathbb{G }}$ be the set of all irreducible and unitary representations of $\mathbb{G }$. Let $\pi \in \hat{\mathbb{G }}$. Then $\pi$ is finite dimensional. Let $a_\pi$ be the degree of $\pi$ and let ${\mathcal{H}_\pi }$ be its representation space. We denote the inner product and the norm in ${\mathcal{H}_\pi }$ by $(\ ,\ )_{{\mathcal{H}_\pi }}$ and $\Vert \ \Vert _{{\mathcal{H}_\pi }}$ respectively. Let $\{\psi _1,\psi _2,\dots ,\psi _{a_\pi }\}$ be an orthonormal basis for ${\mathcal{H}_\pi }$. Then for all $j,k=1,2,\dots ,$ we define $\pi _{jk}$ by

$$\begin{aligned} \pi _{jk}(g)=\sqrt{a_\pi }\left( \pi (g)\psi _k,\psi _j\right) _{{\mathcal{H}_\pi }} \end{aligned}$$

Theorem 3.2

(Peter-Weyl Theorem) $\{\pi _{jk}:\ \pi \in \hat{\mathbb{G }},\ j,k=1,2,\dots ,a_\pi \}$ forms an orthonormal basis for $L^2(\mathbb{G })$.

By Peter-Weyl theorem, every $f\in L^2(\mathbb{G })$ can be expressed as

$$\begin{aligned} f=\sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi }\left( f,\pi _{jk}\right) _{L^2(\mathbb{G })}\pi _{jk}. \end{aligned}$$

have the following Plancheral theorem.

Theorem 3.3

For all $f\in L^2(\mathbb{G })$,

$$\begin{aligned} \Vert f\Vert _{L^2(\mathbb{G })}=\left\{ \sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi }|\left( f,\pi _{jk}\right) _{L^2(\mathbb{G })}|^2\right\} ^{1/2}. \end{aligned}$$

Let $\sigma$ be a measurable function on $\mathbb{G }\times \hat{\mathbb{G }}\times \mathbb{N }\times \mathbb{N }$. For all measurable functions $f$ on $\mathbb{G }$, we define $T_\sigma f$ on $\mathbb{G }$ formally by

$$\begin{aligned} \left( T_\sigma f\right) (g)=\sum _{\pi \in \hat{\mathbb{G }}} \sum _{j,k=1}^{a_\pi }\sigma (g,\pi ,j,k)\left( f,\pi _{jk}\right) _{L^2(\mathbb{G })}\pi _{jk}(g),\quad g\in \mathbb{G }. \end{aligned}$$

$T_\sigma$ is called the pseudo-differential operator on $\mathbb{G }$ corresponding to the symbol $\sigma$. For pseudo-differential operators on Euclidean spaces see for example [15, 19].

Let $L^2(\mathbb{G }\times \hat{\mathbb{G }}\times \mathbb{N }\times \mathbb{N })$ be the space all measurable functions $\sigma$ on ${{\mathbb{G }\times \hat{\mathbb{G }}\times \mathbb{N }\times \mathbb{N }}}$ for which

$$\begin{aligned} \Vert \sigma \Vert _{L^2({{\mathbb{G }\times \hat{\mathbb{G }}\times \mathbb{N }\times \mathbb{N }}})}=\left\{ \sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi }\int \limits _{\mathbb{G }}\left| \sigma (g,\pi ,j,k)\pi _{jk}(g)\right| ^2\,d\mu (g)\right\} ^{1/2}<\infty . \end{aligned}$$

We have the following result on the $L^2$-boundedness of pseudo-differential operators on $\mathbb{G }$.

Theorem 3.4

Let $\sigma \in L^2({{\mathbb{G }\times \hat{\mathbb{G }}\times \mathbb{N }\times \mathbb{N }}}).$ Then $T_\sigma :L^2(\mathbb{G })\rightarrow L^2(\mathbb{G })$ is a bounded operator and

$$\begin{aligned} \Vert T_\sigma \Vert _*\le \Vert \sigma \Vert _{L^2({{\mathbb{G }\times \hat{\mathbb{G }}\times \mathbb{N }\times \mathbb{N }}})}, \end{aligned}$$

where $\Vert \ \Vert _*$ is the norm in the $C^*$-algebra of all bounded linear operators from $L^2(\mathbb{G })$ into $L^2(\mathbb{G })$.

Proof

Let $f\in L^2(\mathbb{G })$. By Minkowski’s inequality and the Schwarz inequality, we have

$$\begin{aligned}&\Vert T_\sigma f\Vert _{L^2(\mathbb{G })} \\&\quad = \left( \int \limits _\mathbb{G }|(T_\sigma f)(g)|^2\,d\mu (g)\right) ^{1/2}\\&\quad =\left( \int \limits _{\mathbb{G }}\left| \sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi }\sigma (g,\pi ,j,k)\left( f,\pi _{jk}\right) _{L^2(\mathbb{G })}\pi _{jk}(g)\right| ^2\,d\mu (g)\right) ^{1/2}\\&\quad \le \sum _{\pi \in \hat{\mathbb{G }}}\left( \int \limits _{\mathbb{G }}\left| \sum _{j,k=1}^{a_\pi }\sigma (g,\pi ,j,k)\left( f,\pi _{jk}\right) _{L^2(\mathbb{G })}\pi _{jk}(g)\right| ^2\,d\mu (g)\right) ^{1/2}\\&\quad \!\le \!\sum _{\pi \in \hat{\mathbb{G }}}\left( \int \limits _{\mathbb{G }} \left( \sum _{j,k=1}^{a_\pi }\left| \left( f,\pi _{jk}\right) _{L^2(\mathbb{G })}\right| ^2\right) \left( \sum _{j,k=1}^{a_\pi }\left| \sigma (g,\pi ,j,k)\pi _{jk}(g)\right| ^2\right) \,d\mu (g)\right) ^{1/2}\\&\quad =\sum _{\pi \in \hat{\mathbb{G }}} \left( \sum _{j,k=1}^{a_\pi }\left| \left( f,\pi _{jk}\right) _{L^2(\mathbb{G })}\right| ^2\right) ^{1/2}\left( \int \limits _{\mathbb{G }}\sum _{j,k=1}^{a_\pi }\left| \sigma (g,\pi ,j,k)\pi _{jk}(g)\right| ^2\,d\mu (g)\right) ^{1/2}\\ \!&\quad \!\le \!\left( \,\sum _{\pi \in \hat{\mathbb{G }}} \sum _{j,k=1}^{a_\pi }\left| \left( \!f,\pi _{jk}\right) _{L^2(\mathbb{G })}\right| ^2\right) ^{1/2}\left( \sum _{\pi \in \hat{\mathbb{G }}}\int \limits _{\mathbb{G }}\sum _{j,k=1}^{a_\pi }\left| \sigma (g,\pi ,j,k)\pi _{jk}(g)\right| ^2\,d\mu (g)\right) ^{1/2}\\&\quad =\Vert f\Vert _{L^2(\mathbb{G })}\Vert \sigma \Vert _{L^2({{\mathbb{G }\times \hat{\mathbb{G }}\times \mathbb{N }\times \mathbb{N }}})}. \end{aligned}$$

$\square$

Now we are ready to give a necessary and sufficient condition on $\sigma$ to guarantee Hilbert-Schmidt properties of $T_\sigma$.

Theorem 3.5

Let $\sigma$ be a measurable function on ${{\mathbb{G }\times \hat{\mathbb{G }}\times \mathbb{N }\times \mathbb{N }}}$. Then $T_\sigma$ is a Hilbert-Schmidt operator if and only if $\sigma \in L^2({{\mathbb{G }\times \hat{\mathbb{G }}\times \mathbb{N }\times \mathbb{N }}})$ and

$$\begin{aligned} \Vert T_\sigma \Vert _{HS}=\Vert \sigma \Vert _{L^{2}({{\mathbb{G }\times \hat{\mathbb{G }}\times \mathbb{N }\times \mathbb{N }}})}. \end{aligned}$$

Proof

For $\tilde{\pi }\in \hat{\mathbb{G }}$ and $j_0,k_0=1,2,\dots ,a_{\tilde{\pi }}$, we have

$$\begin{aligned} \left( T_\sigma \tilde{\pi }_{j_0k_0}\right) (g)&= \sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=0}^{a_\pi }\sigma (g,\pi ,j,k)\left( \tilde{\pi }_{j_0k_0},\pi _{jk}\right) _{L^2(\mathbb{G })}\pi _{jk}(g)\\&= \sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=0}^{a_\pi }\sigma (g,\pi ,j,k)\left( \tilde{\pi }_{j_0k_0},\pi _{jk}\right) _{L^2(\mathbb{G })}\pi _{jk}(g)\\&= \sigma (g,\tilde{\pi },k_0,j_0)\tilde{\pi }_{j_0k_0}(g),\quad g\in \mathbb{G }. \end{aligned}$$

Hence

$$\begin{aligned}&\sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi }\Vert T_\sigma \pi _{jk}\Vert _{L^2(\mathbb{G })}^2\\&\quad =\sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi }\left( \,\int \limits _{\mathbb{G }}|T_\sigma \pi _{jk}(g)|^2\,d\mu (g)\right) \\&\quad =\sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi }\int \limits _{\mathbb{G }}|\sigma (g,\pi ,j,k)\pi _{jk}(g)|^2\,d\mu (g)\\&\quad =\Vert \sigma \Vert _{L^2({{\mathbb{G }\times \hat{\mathbb{G }}\times \mathbb{N }\times \mathbb{N }}})}^2. \end{aligned}$$

and the proof is complete.$\square$

Let $\sigma$ and $\tau$ measurable functions on ${{\mathbb{G }\times \hat{\mathbb{G }}\times \mathbb{N }\times \mathbb{N }}}$. Then we define $\sigma \circledast \tau$ by

$$\begin{aligned}&\sigma \circledast \tau (g,\xi ,l,m)\nonumber \\&\quad =\left( \int \limits _{\mathbb{G }}\tau (w,\xi ,l,m)\xi _{lm}(w)\!\sum _{\pi \in \hat{\mathbb{G }}}\!\sum _{j,k=1}^{a_\pi } \sigma (g,\pi ,j,k)\overline{\pi _{jk}(w)}\pi _{jk}(g)\,d\mu (w)\!\right) (\xi _{lm}(g))^{-1}\nonumber \\ \end{aligned}$$

(3.1)

for all $g\in \mathbb{G }$, $\xi \in \hat{\mathbb{G }}$ and $l,m=1,\dots ,a_\xi$. The following theorem shows that that the composition of two pseudo-differential operators on $\mathbb{G }$ is again a pseudo-differential operator.

Theorem 3.6

Let $\sigma$ and $\tau$ be measurable functions on ${{\mathbb{G }\times \hat{\mathbb{G }}\times \mathbb{N }\times \mathbb{N }}}$. Then

$$\begin{aligned} T_\sigma T_\tau =T_\lambda \end{aligned}$$

where $\lambda =\sigma \circledast \tau .$

Proof

Let $f$ be in $L^2(\mathbb{G })$. Then for all $g\in \mathbb{G }$

$$\begin{aligned}&\left( T_\sigma T_\tau f\right) (g)\nonumber \\&\quad =\sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi }\sigma (g,\pi ,j,k)\left( T_\tau f,\pi _{jk}\right) _{L^2(\mathbb{G })}\pi _{jk}(g)\nonumber \\&\quad =\sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi }\sigma (g,\pi ,j,k)\left( \int \limits _{\mathbb{G }}\left( T_\tau f\right) (w)\overline{\pi _{jk}(w)}\,d\mu (w)\right) \pi _{jk}(g)\nonumber \\&\quad =\sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi }\sigma (g,\pi ,j,k)\left( \int \limits _{\mathbb{G }}\sum _{\xi \in \hat{\mathbb{G }}} \sum _{l,m=1}^{a_\xi }\tau (w,\xi ,l,m)\left( f,\xi _{lm}\right) _{L^2(\mathbb{G })}\xi _{lm}(w)\overline{\pi _{jk}(w)}\,d\mu (w)\right) \pi _{jk}(g)\nonumber \\&\quad =\sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi }\sigma (g,\pi ,j,k)\left( \sum _{\xi \in \hat{\mathbb{G }}} \sum _{l,m=1}^{a_\xi }\left( f,\xi _{lm}\right) _{L^2(\mathbb{G })}\int \limits _{\mathbb{G }}\tau (w,\xi ,l,m)\xi _{lm}(w)\overline{\pi _{jk}(w)}\,d\mu (w)\right) \pi _{jk}(g)\nonumber \\&\quad =\sum _{\xi \in \hat{\mathbb{G }}} \sum _{l,m=1}^{a_\xi }\left( \int \limits _{\mathbb{G }}\tau (w,\xi ,l,m)\xi _{lm}(w)\sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi }\sigma (g,\pi ,j,k)\overline{\pi _{jk}(w)}\pi _{jk}(g)\,d\mu (w)\right) \nonumber \\&\qquad \times (\xi _{lm}(g))^{-1}\left( f,\xi _{lm}\right) _{L^2(\mathbb{G })}\xi _{lm}(g)\nonumber \\&\quad =\sum _{\xi \in \hat{\mathbb{G }}} \sum _{l,m=1}^{a_\xi }\lambda (g,\xi ,l,m)\left( f,\xi _{lm}\right) _{L^2(\mathbb{G })}\xi _{lm}(g), \end{aligned}$$

(3.2)

where $\lambda =\sigma \circledast \tau .$ Thus,

$$\begin{aligned} T_\lambda =T_\sigma T_\tau \end{aligned}$$

$\square$

Now we are ready to give a characterization of trace class pseudo-differential operators on $\mathbb{G }$.

Theorem 3.7

Let $\lambda$ be a measurable function in ${{\mathbb{G }\times \hat{\mathbb{G }}\times \mathbb{N }\times \mathbb{N }}}$. Then $T_\lambda \in S_1$ if and only if there exist symbols $\sigma$ and $\tau$ in $L^2({{\mathbb{G }\times \hat{\mathbb{G }}\times \mathbb{N }\times \mathbb{N }}})$ such that

$$\begin{aligned} \lambda =\sigma \circledast \tau , \end{aligned}$$

and

$$\begin{aligned} tr(T_\lambda )&= \sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi }\int \limits _\mathbb{G }\lambda (g,\pi ,j,k)|\pi _{jk}(g)|^2\,d\mu (g)\\ \!&= \!\sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi }\sum _{\xi \in \hat{\mathbb{G }}}\sum _{l,m=1}^{a_\xi } \left( \tau (\cdot ,\xi ,l,m)\xi _{lm},\pi _{jk}\right) _{L^2(\mathbb{G })} \left( \sigma (\cdot ,\pi ,j,k)\pi _{jk} ,\xi _{lm}\right) _{L^2(\mathbb{G })}\!. \end{aligned}$$

Proof

The first part of the theorem follows from Theorem 2.2, Theorem 3.5 and Theorem 3.6.Now we check the absolute convergence of the series

$$\begin{aligned} \sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi }\sum _{\xi \in \hat{\mathbb{G }}}\sum _{l,m=1}^{a_\xi } \left( \tau (\cdot ,\xi ,l,m)\xi _{lm},\pi _{jk}\right) _{L^2(\mathbb{G })} \left( \sigma (\cdot ,\pi ,j,k)\pi _{jk} ,\xi _{lm}\right) _{L^2(\mathbb{G })} \end{aligned}$$

By the Schwarz inequality and Plancheral’s theorem,

$$\begin{aligned}&\sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi }\sum _{\xi \in \hat{\mathbb{G }}}\sum _{l,m=1}^{a_\xi } \left| \left( \tau (\cdot ,\xi ,l,m)\xi _{lm},\pi _{jk}\right) _{L^2(\mathbb{G })}\right| \left| \left( \sigma (\cdot ,\pi ,j,k)\pi _{jk} ,\xi _{lm}\right) _{L^2(\mathbb{G })}\right| \\&\quad \le \left\{ \sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi }\sum _{\xi \in \hat{\mathbb{G }}}\sum _{l,m=1}^{a_\xi } \left| \left( \tau (\cdot ,\xi ,l,m)\xi _{lm},\pi _{jk}\right) _{L^2(\mathbb{G })}\right| ^2\right\} ^{1/2}\\&\left\{ \sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi }\sum _{\xi \in \hat{\mathbb{G }}}\sum _{l,m=1}^{a_\xi } \left| \left( \sigma (\cdot ,\pi ,j,k)\pi _{jk} ,\xi _{lm}\right) _{L^2(\mathbb{G })}\right| ^2\right\} ^{1/2} \\&\quad =\left\{ \sum _{\xi \in \hat{\mathbb{G }}}\sum _{l,m=1}^{a_\xi } \int \limits _{\mathbb{G }}\left| \tau (g,\xi ,l,m)\xi _{lm}(g)\right| ^2\,d\mu (g)\right\} ^{1/2}\\&\left\{ \sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\xi } \int \limits _{\mathbb{G }}\left| \sigma (g,\pi ,j,k)\pi _{jk}(g)\right| ^2\,d\mu (g)\right\} ^{1/2} \\&\quad =\Vert \sigma \Vert _{L^2({{\mathbb{G }\times \hat{\mathbb{G }}\times \mathbb{N }\times \mathbb{N }}})}\Vert \tau \Vert _{L^2({{\mathbb{G }\times \hat{\mathbb{G }}\times \mathbb{N }\times \mathbb{N }}})}<\infty . \end{aligned}$$

Since $\bigcup _{\pi \in \hat{\mathbb{G }}}\{\pi _{jk}:\ j,k=1,2,\dots ,a_\pi \}$ forms an orthonormal basis for $L^2(\mathbb{G })$, it follows that

$$\begin{aligned}&tr(T_\lambda )=\sum _{\xi \in \hat{\mathbb{G }}}\sum _{l,m=1}^{a_\xi }\left( T_\lambda \xi _{lm},\xi _{lm}\right) _{L^2(\mathbb{G })}\\&\quad =\sum _{\xi \in \hat{\mathbb{G }}}\sum _{l,m=1}^{a_\xi }\int \limits _{\mathbb{G }}\lambda (g,\xi ,l,m)\xi _{lm}(g)\overline{\xi _{lm}(g)}\,d\mu (g)\\ \!&\quad =\!\!\sum _{\xi \in \hat{\mathbb{G }}}\!\sum _{l,m=1}^{a_\xi }\int \limits _{\mathbb{G }} \left( \int \limits _{\mathbb{G }}\tau (w,\xi ,l,m)\xi _{lm}(w)\sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi } \sigma (g,\!\pi \!,j,k)\overline{\pi _{jk}(w)}\pi _{jk}(g)\,d\mu (w)\!\right) \\&\qquad \times \overline{\xi _{lm}(g)}\,d\mu (g)\\&\quad =\sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi }\sum _{\xi \in \hat{\mathbb{G }}}\sum _{l,m=1}^{a_\xi }\int \limits _{\mathbb{G }} \tau (w,\xi ,l,m)\xi _{lm}(w)\overline{\pi _{jk}(w)}\,d\mu (w) \int \limits _{\mathbb{G }}\sigma (g,\pi ,j,k)\pi _{jk}(g)\\&\qquad \times \overline{\xi _{lm}(g)}\,d\mu (g)\\&\quad =\sum _{\pi \in \hat{\mathbb{G }}}\sum _{j,k=1}^{a_\pi }\sum _{\xi \in \hat{\mathbb{G }}}\sum _{l,m=1}^{a_\xi } \left( \tau (\cdot ,\xi ,l,m)\xi _{lm},\pi _{jk}\right) _{L^2(\mathbb{G })} \left( \sigma (\cdot ,\pi ,j,k)\pi _{jk} ,\xi _{lm}\right) _{L^2(\mathbb{G })}. \end{aligned}$$

$\square$

We can now look at the special case when $\mathbb{G }=\mathbb{S }^1.$

Example 3.8

Let ${\mathbb{S }^1}$ be the unit circle centered at the origin. For all $n\in \mathbb{Z }$, we define $\pi _n:\mathbb{S }^1\rightarrow U(1)$ by

$$\begin{aligned} \pi _n(\theta )=(2\pi )^{-1/2}e^{in\theta },\quad \theta \in [-\pi ,\pi ]. \end{aligned}$$

Then

$$\begin{aligned} \widehat{\mathbb{S }^1}=\{\pi _n:\ n\in \mathbb{Z }\}, \end{aligned}$$

and $\widehat{\mathbb{S }}\simeq \mathbb{Z }.$ Hence for any symbol $\sigma$ on $\mathbb{S }^1\times \mathbb{Z }$ we can define pseudo-differential operator $T_\sigma$ on $L^2(\mathbb{S }^1)$ by

$$\begin{aligned} \left( T_{\sigma }f\right) (\theta )=\sum _{n\in \mathbb{Z }}\sigma (\theta ,n)(f,\pi _n)_{L^2(\mathbb{S }^1)}\pi _n(\theta ),\quad f\in L^2(\mathbb{S }^1),\ \theta \in [-\pi ,\pi ]. \end{aligned}$$

Hence,

$$\begin{aligned} \left( T_\sigma f\right) (\theta )=(2\pi )^{-1}\sum _{n\in \mathbb{Z }}\,\int \limits _{-\pi }^{\pi }e^{in(\theta -\varphi )}\sigma (n,\theta )f(\varphi )\,d\varphi . \end{aligned}$$