# Dixmier traces, Wodzicki residues, and determinants on compact Lie groups:
the paradigm of the global quantisation

Duván Cardona  Duván Cardona: Department of Mathematics: Analysis, Logic and
Discrete Mathematics Ghent University, Belgium E-mail address
<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Julio Delgado  Julio
Delgado: Departmento de Matemáticas Universidad del Valle Cali-Colombia E-mail
address<EMAIL_ADDRESS>and  Michael Ruzhansky  Michael
Ruzhansky: Department of Mathematics: Analysis, Logic and Discrete Mathematics
Ghent University, Belgium and School of Mathematical Sciences Queen Mary
University of London United Kingdom E-mail address<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

By following the paradigm of the global quantisation, instead of the analysis
under changes of coordinates, in this work we establish a global analysis for
the explicit computation of the Dixmier trace and the Wodzicki residue of
(elliptic and subelliptic) pseudo-differential operators on compact Lie
groups. The regularised determinant for the Dixmier trace is also computed. We
obtain these formulae in terms of the global symbol of the corresponding
operators. In particular, our approach links the Dixmier trace and Wodzicki
residue to the representation theory of the group. Although we start by
analysing the case of compact Lie groups, we also compute the Dixmier trace
and its regularised determinant on arbitrary closed manifolds $M$, for the
class of invariant pseudo-differential operators in terms of their matrix-
valued symbols. This analysis includes e.g. the family of positive and
elliptic pseudo-differential operators on $M$.

###### Key words and phrases:

Dixmier trace, trace formula, determinants, pseudo-differential operators

###### 2020 Mathematics Subject Classification:

43A15, 43A22; Secondary 22E25, 43A80

The authors are supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and
Partial Differential Equations and by the Methusalem programme of the Ghent
University Special Research Fund (BOF) (Grant number 01M01021). Julio Delgado
is also supported by Vice. Inv. Universidad del Valle Grant CI71281. Michael
Ruzhansky is also supported by EPSRC grant EP/R003025/2.

###### Contents

  1. 1 Introduction
    1. 1.1 Outline
    2. 1.2 Main results
  2. 2 Preliminaries
    1. 2.1 Global Hörmander classes of pseudo-differential operators
    2. 2.2 Hörmander classes in the presence of a sub-Riemannian structure
    3. 2.3 Determinants on Hilbert spaces
    4. 2.4 Determinant of invariant operators on Hilbert spaces
    5. 2.5 The Dixmier ideal on Hilbert spaces
    6. 2.6 Connes equivalence theorem
  3. 3 Dixmier traces and Wodzicki residues on compact Lie groups
  4. 4 Final remarks on compact manifolds

##  1\. Introduction

###  1.1. Outline

In this work we compute the Wodzicki-Guillemin residue and the Dixmier trace
of pseudo-differential operators on compact Lie groups in terms of their
global symbols, or equivalently, in terms of the representation theory of
these groups. We prove our geometric/spectral formulae in the context of
pseudo-differential operators associated to Riemannian and sub-Riemannian
structures on compact Lie groups [RT10, CR20]. The validity of such results
can be conjectured from the techniques developed in [LNP16, PS07] where the
Wodzicki residue was studied by using a twisted version of the global-symbolic
calculus on the torus developed by the third author with Turunen [RT10]. See
also [CDC20]. Indeed, the theory in [RT10], can be understood as a
discretisation of the algebraic formalism by Connes and Moscovici [CM95] in
the setting of Lie groups, making that theory suitable for the global analysis
of residues, determinants and traces.

As it was pointed out by Connes (cf. [Con94]), in algebraic quantum field
theory, in order to write down an action in operator language, a functional
that replaces integration is required. For the Yang-Mills theory, that is the
Dixmier trace $\textnormal{\bf Tr}_{\omega}$, which in contrast to the usual
trace, vanishes on the ideal of trace class operators, and is defined in the
ideal $\mathcal{L}^{(1,\infty)}$ of the compact operators $T$ such that the
partial sums of their singular values diverge logarithmically. On the other
hand, the Wodzicki (or non-commutative) residue, is the only extension of the
Dixmier trace on the algebra of classical pseudo-differential operators, in
view of a fundamental result due to Connes [Con94]. These two traces, are of
relevant interests in several branches of mathematics such as differential
geometry, spectral theory, mathematical physics, and non-commutative geometry
(c.f. [Con94]). A reason for this is that they allow to study specific
topological/geometric invariants, where very specific elliptic operators –such
as the Dirac operator– have been shown to contain very important information
relevant for classification problems, see Lesch [Les99], Paycha [Pay12],
Sukochev and Usachev [SU16], Scott [Sco10], Grubb and Schrohe [GS04], Fedosov,
Golse, Leichtnam, and Schrohe [FGLS96] and references therein, just to mention
a few.

According to the theory developed by Hörmander [Hör85], to any pseudo-
differential operator $A$ acting on $C^{\infty}(M),$ with $M$ a closed
manifold $M$ (i.e. compact and without boundary) one can associate a symbol,
defined by local coordinate systems, which is a function on the cotangent
bundle $T^{*}M.$ However, some information about the operator (such as its
lower terms) can be lost from the geometric point of view. On the other hand,
it was described by the third author and Turunen in [RT10], that if $M=G$ is a
compact Lie group, one can give a global definition of symbol for the
Hörmander classes, and that these symbols can be realised as matrix-valued
functions of the phase space $G\times\widehat{G},$ where $\widehat{G}$ is the
unitary dual of $G$, that is the set of equivalence classes of continuous
irreducible unitary representations on the group.

In this work we address the following problems: (1) to compute the Dixmier
trace and the regularised determinant associated with it, for pseudo-
differential operators when $M=G$ is a compact Lie group. The reason of
considering the case of a compact Lie group is to exploit the symmetries of
the group as in [RT10]. Also, by following [CR20], we also consider the
problem of computing the Dixmier trace of subelliptic pseudo-differential
operators. (2) To compute the Wodzicki residue of (elliptic and subelliptic)
pseudo-differential operators on a compact Lie group $G$ in terms of their
matrix-valued symbols or equivalently, in terms of the representation theory
of the group, and (3) to compute the regularised determinant for the Dixmier
trace in the case of invariant pseudo-differential operators on a compact
manifold $M$ in terms of their matrix-valued symbols, by following the
formalism developed in [DR18], where the global symbols are defined in terms
of a global Fourier analysis on $M$ and covers the case of elliptic and
positive pseudo-differential operators on $M.$

###  1.2. Main results

For simplicity, we will discuss briefly our main results in the setting of
positive (elliptic or subelliptic) pseudo-differential operators. However, we
present the general situation without positivity assumptions in Section 3.

By following Hörmander [Hör85], a classical pseudo-differential operator $A$
on a closed manifold $M$ of dimension $n,$ has a symbol

$\sigma^{A}(x,\xi)\sim\sum_{j=0}^{\infty}\sigma^{A}_{m-j}(x,\xi),\,\,(x,\xi)\in
T^{*}M,$

defined by localisations, and the Wodzicki residue, which measures the
locality of the operator, is given by

$\textnormal{res}\,(A)=\frac{1}{n(2\pi)^{n}}\int\limits_{M}\int\limits_{|\xi|=1}\sigma^{A}_{-n}(x,\xi)\,d\xi\,dx.$
(1.1)

By a fundamental theorem of Connes [Con94], if $A$ has order $-n,$ then
$\textnormal{res}\,(A)$ coincides with the Dixmier trace of $A$, that is:

$\boxed{{\textnormal{ Spectral (global) information }=\textnormal{\bf
Tr}_{\omega}(A)=\textnormal{res}\,(A)=\textnormal{ Geometric (local)
information}}}$

Now, if $M$ has symmetries, for instance if $M$ is the torus,
$M=\textnormal{SU}(2),$ or $M=G$ is an arbitrary compact Lie group, the work
of the third author and Turunen [RT10] shown that for describing the Hörmander
classes in [Hör85], is enough to consider the representation theory of $G.$
Then, a natural question is if the Wodzicki residue can be computed from this
algebraic formalism, or equivalently:

$\boxed{{\textnormal{is }\textnormal{res}\,(A)\,\,\textnormal{ information
encoded by the representation theory of }G\textnormal{?}}}$ (1.2)

In this work we answer (1.2) in the affirmative, by proving the validity of
the formula

$\textnormal{res}(A)=\int\limits_{G}\|\sigma_{-n}(x,[\xi])\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}dx,$
(1.3)

for a positive pseudo-differential operator $A$ of order $-n,$ where the full
symbol $\sigma_{-n}(x,[\xi]),$ $(x,[\xi])\in G\times\widehat{G},$ is the
component of order $-n,$ of the matrix-valued symbol $\sigma_{A}(x,[\xi])$ of
$A,$ which is a global mapping on the non-commutative phase space
$G\times\widehat{G}$. Here, the $\mathcal{L}^{(1,\infty)}$-norm of the symbol
is given by

$\|\sigma_{-n}(x,[\xi])\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}:=\lim_{N\rightarrow\infty}\frac{1}{\log
N}\sum_{[\xi]:\langle\xi\rangle\leq
N}d_{\xi}\textnormal{{Tr}}(|\sigma_{-n}(x,[\xi])|),$

where the sequence $\langle\xi\rangle,$ $[\xi]\in\widehat{G},$ is formed by
the eigenvalues of the operator $(1+\mathcal{L}_{G})^{\frac{1}{2}},$ with
$\mathcal{L}_{G}=-\sum_{i=1}^{n}X_{j}^{2}$ being the positive Laplacian on
$G,$ see Remark 2.2 for details. Moreover, we compute the Wodzicki residue for
classical pseudo-differential operators in Corollary 3.11 without the
assumption of the positivity of the operator.

The residue formula will be obtained for classical operators as a consequence
of a more general formula (see Corollary 3.5) that we prove in the setting of
subelliptic operators on compact Lie groups [CR20]. Indeed, in Corollary 3.6,
for a positive subelliptic operator $A,$ we prove the formula

$\textnormal{\bf{Tr}}_{\omega}(A)=\int\limits_{G}\|\sigma_{-Q}(x,[\xi])\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}dx=\lim_{N\rightarrow\infty}\frac{1}{\log(N)}\sum_{j=1}^{N}s_{j}(A),$
(1.4)

where $s_{j}(A)$ denotes the sequence of singular values of $A$ with
multiplicities taken into account. We refer the reader to Theorem 3.9 for the
Dixmier trace formula of $A$ without the positivity assumption. In (1.4), $Q$
denotes the Hausdorff dimension associated to a positive sub-Laplacian
$\mathcal{L}=-\sum_{j=1}^{k}X_{j}^{2},$ associated to a Hörmander system of
vector-fields $\\{X_{j}\\}_{j=1}^{k}$ (see Subsection 2.2 for details), and
$A$ is a positive operator in the Hörmander classes associated to
$\mathcal{L},$ see Subsection 2.2 or [CR20] for details. Let us remark that
(1.4) is a Lidskii type formula, that relates the Dixmier trace with the
eigenvalues of the operator. An analogous of the Lidskii formula can be
obtained in the setting of Dixmier traces following the recent work of
Sukochev, Zanin et al [GG17], [SSZ15], [SZ14b], [GS07], [SZ14a].

We conclude the introduction by focusing our attention on two aspects of the
analysis on Lie groups that are regarded as quite far apart, and that are
related with the validity of (1.3). One comes from the theories in [RT10,
CR20] which show that the analysis of pseudo-differential operators using
global symbols can be effectively treated when working on compact Lie groups.
Indeed, different techniques regarding the representation theory of a compact
Lie group (where the unitary dual is a discrete set) allow to use
generalisations of the Fourier analysis on the torus, or on
$\textnormal{SU(2)}.$ In this framework Lidskii type formulae are used to
localise the spectrum of an operator and to establish trace formulae in terms
of the matrix-valued symbol of an operator (c.f. [Del10a, Del10b, DR14a,
DR14b, DR17, DR18]). Another aspect, from the point of view of the twisted
versions of the periodic analysis in [RT10], or equivalently, from the twisted
structures arising in the non-commutative geometry, e.g. by deformation of the
torus, or by deformation of $\textnormal{SU}(2),$ shows that one can
generalise Wodzicki type results as it was firstly observed in [LNP16] using
the notion of a global symbol (with predecessor works on the subject as in
[FK13]). Extension of the Dixmier trace in a more general setting of Banach
spaces is available from the works of Pietsch [Pie12]. For a recent survey on
the theory of traces and their history we refer the reader to Pietsch [Pie14].

The manuscript is organised as follows: In Section 2 we first recall basic
notions of pseudodifferential calculus on compact Lie groups, subelliptic
classes, determinants on Hilbert spaces and the Dixmier ideal. In Section 3 we
study Dixmier traces and regularised determinants. We present there the main
results of this work. For instance, we prove the validity of (1.3) in its more
general form for subelliptic operators in Theorem 3.9, and consequently for
classical elliptic pseudo-differential operators in Corollary 3.10. In
Corollary 3.11 we compute a formula for the Wodzicki residue of elliptic and
classical pseudo-differential operators in terms of their global symbols.
Finally, in Section 4 we give some final remarks about the regularised
determinant for the Dixmier trace on arbitrary compact manifolds.

##  2\. Preliminaries

In this section we recall some basics of the pseudodifferential calculus on
compact Lie groups, the subelliptic Hörmander classes recently introduced in
[CR20], determinants on Hilbert spaces and the Dixmier ideal.

###  2.1. Global Hörmander classes of pseudo-differential operators

First, let us record the notion of the unitary dual $\widehat{G}$ of a compact
Lie group $G.$ So, let us assume that $\xi$ is a continuous, unitary and
irreducible representation of $G,$ that is

  * •

$\xi\in\textnormal{Hom}(G,\textnormal{U}(H_{\xi})),$ for some finite-
dimensional vector space $H_{\xi}\cong\mathbb{C}^{d_{\xi}},$ i.e.
$\xi(xy)=\xi(x)\xi(y)$ and for the adjoint of $\xi(x),$
$\xi(x)^{*}=\xi(x^{-1}),$ for every $x,y\in G.$

  * •

The map $(x,v)\mapsto\xi(x)v,$ from $G\times H_{\xi}$ into $H_{\xi}$ is
continuous.

  * •

For every $x\in G,$ and $W_{\xi}\subset H_{\xi},$ if $\xi(x)W_{\xi}\subset
W_{\xi},$ then $W_{\xi}=H_{\xi}$ or $W_{\xi}=\emptyset.$

Let $\textnormal{Rep}(G)$ be the set of unitary, continuous and irreducible
representations of $G.$ The relation,

$\xi_{1}\sim\xi_{2}\textnormal{ if and only if, there exists
}\varkappa\in\textnormal{End}(H_{\xi_{1}},H_{\xi_{2}}),\textnormal{ such that
}\varkappa\xi_{1}(x)\varkappa^{-1}=\xi_{2}(x),$

for every $x\in G,$ is an equivalence relation and the unitary dual of $G,$
denoted by $\widehat{G}$ is defined via

$\widehat{G}:={\textnormal{Rep}(G)}/{\sim}.$

The compactness of $G$ implies that $\widehat{G}$ is a discrete set. By a
suitable changes of basis, we always can assume that every $\xi$ is matrix-
valued and that $H_{\xi}=\mathbb{C}^{d_{\xi}}.$ If a representation $\xi$ is
unitary, then

$\xi(G):=\\{\xi(x):x\in G\\}$

is a subgroup (of the group of matrices $\mathbb{C}^{d_{\xi}\times d_{\xi}}$)
which is isomorphic to the original group $G$. Thus the homomorphism $\xi$
allows us to represent the compact Lie group $G$ as a group of matrices. This
is the motivation for the term ‘representation’.

The prototype of differential operators of the first order are vector-fields.
One can associate first differential operators to vector fields using the
exponential mapping as follows. We denote by $\mathfrak{g}$ the Lie algebra of
$G$.

######  Remark 2.1.

Every $X\in\mathfrak{g},$ can be identified with the differential operator
$X:C^{\infty}(G)\rightarrow C^{\infty}(G)$ defined by

$(X_{x}f)(x):=\frac{d}{dt}f(x\exp(tX))|_{t=0},\quad x\in G.$

The Laplacian on a compact Lie group is the model of a differential operator
of second order. Its eigenvalues will be used to define the Hörmander classes
on $G$ (see Remark 2.7 below). It can be defined as follows.

######  Remark 2.2 (The Laplacian on $G$).

Let $G$ be a compact Lie group of dimension $n$ and let
$\textnormal{ad}:\mathfrak{g}\rightarrow\textnormal{End}(\mathfrak{g})$ be the
adjoint representation of its Lie algebra $\mathfrak{g}$. With respect to the
the Killing form

$B(Y_{1},Y_{2}):=\textnormal{Tr}(\textnormal{ad}(Y_{1})\circ\textnormal{ad}(Y_{2})),$

on $\mathfrak{g},$ let us choose an orthonormal basis $X_{1},\cdots,X_{n},$ of
$\mathfrak{g}.$ The positive Laplacian on $G,$ (Casimir element) is defined
via

$\mathcal{L}_{G}=-\sum_{j=1}^{n}X_{j}^{2}.$ (2.1)

The operator $\mathcal{L}_{G}$ has $L^{2}(G)$-discrete spectrum
$\\{\lambda_{[\xi]}\\}_{[\xi]\in\widehat{G}},$ and the eigenvalues of the
(Bessel potential) operator $(1+\mathcal{L}_{G})^{\frac{1}{2}}$ are given by

$\langle\xi\rangle:=(1+\lambda_{[\xi]})^{\frac{1}{2}},$ (2.2)

in view of the spectral mapping theorem. The weight function (2.2) will be
crucial in defining the Hörmander classes of symbols.

######  Remark 2.3.

Let $[\xi]\in\widehat{G},$ and $\xi\in[\xi].$ The entry-functions $\xi_{ij}$
of any unitary representation $\xi=(\xi_{ij})_{i,j=1}^{d_{\xi}}$ span an
eigenspace $\mathcal{H}^{\xi}\subset L^{2}(G)$ of the Laplacian and the
restriction $\mathcal{L}_{G}|_{\mathcal{H}^{\xi}}=\lambda_{[\xi]}I,$ is a
multiple of the identity operator on $\mathcal{H}^{\xi},$ see Theorem 10.3.13
of [RT10]. The number $\lambda_{[\xi]}$ is independent of the choice of
$\xi\in[\xi].$ Consequently, for any $\xi,$ the entry-functions $\xi_{ij}$ are
of $C^{\infty}$-class. Indeed, they are eigenfunctions of the positive Laplace
operator $\mathcal{L}_{G}$, that is
$\mathcal{L}_{G}\xi_{ij}=\lambda_{[\xi]}\xi_{ij}.$

We now follow [RT10, Chapter 10] to introduce the analysis of operators on the
phase space $G\times\widehat{G}.$ Indeed, if $A$ is a continuous linear
operator on $C^{\infty}(G),$ there exists a function

$a:G\times\widehat{G}\rightarrow\cup_{\ell\in\mathbb{N}}\mathbb{C}^{\ell\times\ell},$
(2.3)

such that for every equivalence class $[\xi]\in\widehat{G},$
$a(x,\xi):=a(x,[\xi])\in\mathbb{C}^{d_{\xi}\times d_{\xi}},$ (where $d_{\xi}$
is the dimension of the continuous, unitary and irreducible representation
$\xi:G\rightarrow\textnormal{U}(\mathbb{C}^{d_{\xi}})$) and satisfying

$Af(x)=\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{{Tr}}[\xi(x)a(x,\xi)\widehat{f}(\xi)],\,\,f\in
C^{\infty}(G).$ (2.4)

Here

$\widehat{f}(\xi)\equiv(\mathscr{F}f)(\xi):=\int\limits_{G}f(x)\xi(x)^{*}dx\in\mathbb{C}^{d_{\xi}\times
d_{\xi}},\,\,\,[\xi]\in\widehat{G},$

is the matrix-valued Fourier transform of $f$ at
$\xi=(\xi_{ij})_{i,j=1}^{d_{\xi}},$ and the function $a$ in (2.4) satisfies
the identity,

$a(x,\xi)\equiv\widehat{A}(x,\xi)\equiv\sigma_{A}(x,\xi)=\xi(x)^{*}(A\xi)(x),\,\,A\xi:=(A\xi_{ij})_{i,j=1}^{d_{\xi}},\,\,\,\,[\xi]\in\widehat{G}.$
(2.5)

In general, we refer to the function $a$ as the (global or matrix) symbol of
the operator $A.$

######  Remark 2.4.

We denote by $\mathscr{S}(\widehat{G}):=\mathscr{F}(C^{\infty}(G))$ the
Schwartz space on the unitary dual. Then the Fourier transform on the group
$\mathscr{F}$ is a bijective mapping from $C^{\infty}(G)$ into
$\mathscr{S}(\widehat{G})$ (see [RT10, Page 541]), and in terms of the Fourier
transform we have

$Af(x)=\mathscr{F}^{-1}[a(x,\cdot)(\mathscr{F}{f})\,](x),$

for every $f\in C^{\infty}(G).$ In particular, if $a(x,\xi)=I_{d_{\xi}}$ is
the identity matrix in every representation space, $A\equiv I$ is the identity
operator on $C^{\infty}(G),$ and we recover the Fourier inversion formula

$f(x)=\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{Tr}[\xi(x)\widehat{f}(\xi)],\,\,f\in
C^{\infty}(G).$

To classify symbols in the Hörmander classes developed in [RT10], the notion
of  difference operators on the unitary dual, by endowing $\widehat{G}$ with a
difference structure, is an instrumental tool. By following [RW15], a
difference operator $Q_{\xi}$ of order $k,$ is defined by

$Q_{\xi}\widehat{f}(\xi)=\widehat{qf}(\xi),\,[\xi]\in\widehat{G},$ (2.6)

for all $f\in C^{\infty}(G),$ for some function $q$ vanishing of order $k$ at
the identity $e=e_{G}.$ We will denote by $\textnormal{diff}^{k}(\widehat{G})$
the set of all difference operators of order $k.$ For a fixed smooth function
$q,$ the associated difference operator will be denoted by
$\Delta_{q}:=Q_{\xi}.$ We will choose an admissible collection of difference
operators (see e.g. [RW15]),

$\Delta_{\xi}^{\alpha}:=\Delta_{q_{(1)}}^{\alpha_{1}}\cdots\Delta_{q_{(i)}}^{\alpha_{i}},\,\,\alpha=(\alpha_{j})_{1\leqslant
j\leqslant i},$

where

$\textnormal{rank}\\{\nabla q_{(j)}(e):1\leqslant j\leqslant
i\\}=\textnormal{dim}(G),\textnormal{ and
}\Delta_{q_{(j)}}\in\textnormal{diff}^{1}(\widehat{G}).$

We say that this admissible collection is strongly admissible if

$\bigcap_{j=1}^{i}\\{x\in G:q_{(j)}(x)=0\\}=\\{e_{G}\\}.$

######  Remark 2.5.

A special type of difference operators can be defined by using the unitary
representations of $G.$ Indeed, if $\xi_{0}$ is a fixed irreducible and
unitary representation of $G$, consider the matrix

$\xi_{0}(g)-I_{d_{\xi_{0}}}=[\xi_{0}(g)_{ij}-\delta_{ij}]_{i,j=1}^{d_{\xi}},\,\quad
g\in G.$ (2.7)

Then, we associate to the function
$q_{ij}(g):=\xi_{0}(g)_{ij}-\delta_{ij},\quad g\in G,$ a difference operator
via

$\mathbb{D}_{\xi_{0},i,j}:=\mathscr{F}(\xi_{0}(g)_{ij}-\delta_{ij})\mathscr{F}^{-1}.$
(2.8)

If the representation is fixed we omit the index $\xi_{0}$ so that, from a
sequence
$\mathbb{D}_{1}=\mathbb{D}_{\xi_{0},j_{1},i_{1}},\cdots,\mathbb{D}_{n}=\mathbb{D}_{\xi_{0},j_{n},i_{n}}$
of operators of this type we define
$\mathbb{D}^{\alpha}=\mathbb{D}_{1}^{\alpha_{1}}\cdots\mathbb{D}^{\alpha_{n}}_{n}$,
where $\alpha\in\mathbb{N}^{n}$.

######  Remark 2.6 (Leibniz rule for difference operators).

The difference structure on the unitary dual $\widehat{G},$ induced by the
difference operators acting on the momentum variable $[\xi]\in\widehat{G},$
implies the following Leibniz rule

$\displaystyle\Delta_{\xi}^{\alpha}(a_{1}a_{2})(x_{0},\xi)=\sum_{|\gamma|,|\varepsilon|\leqslant|\alpha|\leqslant|\gamma|+|\varepsilon|}C_{\varepsilon,\gamma}(\Delta_{\xi}^{\gamma}a_{1})(x_{0},\xi)(\Delta_{\xi}^{\varepsilon}a_{2})(x_{0},\xi),\quad(x_{0},[\xi])\in
G\times\widehat{G},$

for $a_{1},a_{2}\in C^{\infty}(G,{S}^{\prime}(\widehat{G})).$ For more details
we refer the reader to [RT10].

######  Remark 2.7 (Hörmander classes on Lie groups).

If $A\in\Psi^{m}_{\rho,\delta}(G,\textnormal{loc}),$ $\rho\geqslant 1-\delta,$
the matrix-valued symbol $\sigma_{A}$ of $A$ satisfies (see [RT10, RTW14]),

$\|{X}_{x}^{\beta}\Delta_{\xi}^{\gamma}\sigma_{A}(x,\xi)\|_{\textnormal{op}}\leqslant
C_{\alpha,\beta}\langle\xi\rangle^{m-\rho|\gamma|+\delta|\beta|}$ (2.9)

for all $\beta$ and $\gamma$ multi-indices and all $(x,[\xi])\in
G\times\widehat{G}$. Now, if $0\leqslant\delta,\rho\leqslant 1,$ we say that
$\sigma_{A}\in{S}^{m}_{\rho,\delta}(G\times\widehat{G}),$ if the global symbol
inequalities (2.9) hold true. So, for
$\sigma_{A}\in{S}^{m}_{\rho,\delta}(G\times\widehat{G})$ we write
$A\in\Psi^{m}_{\rho,\delta}(G)\equiv\textnormal{Op}({S}^{m}_{\rho,\delta}(G\times\widehat{G})).$
Observe that

$\textnormal{Op}({S}^{m}_{\rho,\delta}(G\times\widehat{G}))=\Psi^{m}_{\rho,\delta}(G,\textnormal{loc}),\,\,\,0\leqslant\delta<\rho\leqslant
1,\,\rho\geqslant 1-\delta.$

An important type of pseudo-differential operators are the elliptic operators.
They are invertible modulo smoothing operators as we discuss in the following
remark.

######  Remark 2.8.

For $m\in\mathbb{R},$ and $0\leqslant\delta<\rho\leqslant 1,$ a symbol
$a=a(x,\xi)\in{S}^{m}_{\rho,\delta}(G\times\widehat{G})$ is elliptic, if
$a(x,\xi)$ is invertible for every $(x,[\xi])\in G\times\widehat{G},$ (with
the exception of finitely many equivalence classes $[\xi]$) and if it
satisfies

$\|a(x,\xi)^{-1}\|_{\textnormal{op}}\leq
C\langle\xi\rangle^{-m},\,\,\,\,\langle\xi\rangle\geq N,$ (2.10)

for some $N\in\mathbb{N}$ large enough. In this case we say that the pseudo-
differential operator $A$ is elliptic of order $m.$ The crucial property of
elliptic operators is that they are of Fredholm type on $L^{2}(G).$ Indeed, by
following [RT10], if $0\leq\delta<\rho\leq 1,$ there exists
$B\in{S}^{-m}_{\rho,\delta}(G\times\widehat{G}),$ such that

$R_{1}:=AB-I,R_{2}:=BA-I\in{S}^{-\infty}(G\times\widehat{G}):=\bigcap_{m\in\mathbb{R}}{S}^{m}_{\rho,\delta}(G\times\widehat{G}).$

Indeed, the operators $R_{1}$ and $R_{2}$ are compact on $L^{2}(G).$ This
holds in general for closed manifolds but the additional condition $\rho\geq
1-\delta$ is required.

###  2.2. Hörmander classes in the presence of a sub-Riemannian structure

The Hörmander classes in [RT10] are defined using the spectrum of the
Laplacian. In [CR20], these classes were extended by using as measure of decay
the spectrum of the sub-Laplacian. This generalisation is a useful tool when
studying subelliptic (analytic and spectral) problems as it was extensively
developed in [CR20]. To introduce this calculus let us fix the necessary
notations.

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}.$ Under the
identification $\mathfrak{g}\simeq T_{e_{G}}G,$ where $e_{G}$ is the identity
element of $G,$ let us consider a system of $C^{\infty}$-vector fields
$X=\\{X_{1},\cdots,X_{k}\\}\in\mathfrak{g}$. For all
$I=(i_{1},\cdots,i_{\omega})\in\\{1,2,\cdots,k\\}^{\omega},$ of length
$\omega\geqslant 2,$ denote

$X_{I}:=[X_{i_{1}},[X_{i_{2}},\cdots[X_{i_{\omega-1}},X_{i_{\omega}}]\cdots]],$

and for $\omega=1,$ $I=(i),$ $X_{I}:=X_{i}.$ Let $V_{\omega}$ be the subspace
generated by the set $\\{X_{I}:|I|\leqslant\omega\\}.$ That $X$ satisfies the
Hörmander condition, means that there exists $\kappa^{\prime}\in\mathbb{N}$
such that $V_{\kappa^{\prime}}=\mathfrak{g}.$ Certainly, we consider the
smallest $\kappa^{\prime}$ with this property and we denote it by $\kappa$
which will be later called the step of the system $X.$ We also say that $X$
satisfies the Hörmander condition of order $\kappa.$ Note that the sum of
squares

$\mathcal{L}\equiv\mathcal{L}_{X}:=-(X_{1}^{2}+\cdots+X_{k}^{2}),$

is a subelliptic operator which by following the usual nomenclature is called
the subelliptic Laplacian associated with the family $X.$ For short we refer
to $\mathcal{L}$ as the sub-Laplacian.

A central notion in the analysis of the sub-Laplacian is that of the Hausdorff
dimension, in this case, associated to $\mathcal{L}$. Indeed, for all $x\in
G,$ denote by $H_{x}^{\omega}G$ the subspace of the tangent space $T_{x}G$
generated by the $X_{i}$’s and all the Lie brackets

$[X_{j_{1}},X_{j_{2}}],[X_{j_{1}},[X_{j_{2}},X_{j_{3}}]],\cdots,[X_{j_{1}},[X_{j_{2}},[X_{j_{3}},\cdots,X_{j_{\omega}}]]],$

with $\omega\leqslant\kappa.$ The Hörmander condition can be stated as
$H_{x}^{\kappa}G=T_{x}G,$ $x\in G.$ We have the filtration

$H_{x}^{1}G\subset H_{x}^{2}G\subset H_{x}^{3}G\subset\cdots\subset
H_{x}^{\kappa-1}G\subset H_{x}^{\kappa}G=T_{x}G,\,\,x\in G.$

In our case, the dimension of every $H_{x}^{\omega}G$ does not depend on $x$
and we write $\dim H^{\omega}G:=\dim H_{x}^{\omega}G,$ for any $x\in G.$ So,
the Hausdorff dimension can be defined as (see e.g. [HK16, p. 6]),

$Q:=\dim(H^{1}G)+\sum_{i=1}^{\kappa-1}(i+1)(\dim H^{i+1}G-\dim H^{i}G).$
(2.11)

We will use the notation $\widehat{\mathcal{M}}$ for the matrix-valued symbol
of the operator $\mathcal{M}:=(1+\mathcal{L})^{\frac{1}{2}},$ and for every
$[\xi]\in\widehat{G},$ and $s\in\mathbb{R},$

$\widehat{\mathcal{M}}(\xi)^{s}:=\textnormal{diag}[(1+\nu_{ii}(\xi)^{2})^{\frac{s}{2}}]_{1\leqslant
i\leqslant d_{\xi}},$

where
$\widehat{\mathcal{L}}(\xi)=:\textnormal{diag}[\nu_{ii}(\xi)^{2}]_{1\leqslant
i\leqslant d_{\xi}}$ is the symbol of the sub-Laplacian $\mathcal{L}$ at
$[\xi].$

In order to introduce the Hörmander classes associated to the sub-Laplacian
$\mathcal{L},$ we require a suitable class of differential operators
$\partial_{X}^{(\beta)}$, compatible with the difference operators and the
remainders in the Taylor expansion formula on compact Lie groups. We introduce
such a family in the following lemma (see Lemma 4.2 of [CR20]).

######  Lemma 2.9 (Global Taylor Series on compact Lie groups).

Let $G$ be a compact Lie group of dimension $n.$ Let us consider an strongly
admissible admissible collection of difference operators
$\mathfrak{D}=\\{\Delta_{q_{(j)}}\\}_{1\leqslant j\leqslant n}$, which means
that

$\textnormal{rank}\\{\nabla q_{(j)}(e):1\leqslant j\leqslant
n\\}=n,\,\,\,\bigcap_{j=1}^{n}\\{x\in G:q_{(j)}(x)=0\\}=\\{e_{G}\\}.$

Then there exists a basis
$X_{\mathfrak{D}}=\\{X_{1,\mathfrak{D}},\cdots,X_{n,\mathfrak{D}}\\}$ of
$\mathfrak{g},$ such that
$X_{j,\mathfrak{D}}q_{(k)}(\cdot^{-1})(e_{G})=\delta_{jk}.$ Moreover, by using
the multi-index notation
$\partial_{X}^{(\beta)}=\partial_{X_{i,\mathfrak{D}}}^{\beta_{1}}\cdots\partial_{X_{n,\mathfrak{D}}}^{\beta_{n}},$
$\beta\in\mathbb{N}_{0}^{n},$ where

$\partial_{X_{i,\mathfrak{D}}}f(x):=\frac{d}{dt}f(x\exp(tX_{i,\mathfrak{D}}))|_{t=0},\,\,f\in
C^{\infty}(G),$

and denoting for every $f\in C^{\infty}(G)$

$R_{x,N}^{f}(y):=f(xy)-\sum_{|\alpha|<N}q_{(1)}^{\alpha_{1}}(y^{-1})\cdots
q_{(n)}^{\alpha_{n}}(y^{-1})\partial_{X}^{(\alpha)}f(x),$

we have that

$|R_{x,N}^{f}(y)|\leqslant C|y|^{N}\max_{|\alpha|\leqslant
N}\|\partial_{X}^{(\alpha)}f\|_{L^{\infty}(G)}.$

The constant $C>0$ is dependent on $N,$ $G$ and $\mathfrak{D},$ but not on
$f\in C^{\infty}(G).$ Also, we have that
$\partial_{X}^{(\beta)}|_{x_{1}=x}R_{x_{1},N}^{f}=R_{x,N}^{\partial_{X}^{(\beta)}f}$
and

$|\partial_{X}^{(\beta)}|_{y_{1}=y}R_{x,N}^{f}(y_{1})|\leqslant
C|y|^{N-|\beta|}\max_{|\alpha|\leqslant
N-|\beta|}\|\partial_{X}^{(\alpha+\beta)}f\|_{L^{\infty}(G)},$

provided that $|\beta|\leqslant N.$

Now, we introduce the Hörmander classes for the sub-Laplacian as developed in
[CR20].

######  Definition 2.10 (Subelliptic Hörmander classes).

Let $G$ be a compact Lie group and let $0\leqslant\delta,\rho\leqslant 1.$ Let
us consider a sub-Laplacian $\mathcal{L}=-(X_{1}^{2}+\cdots+X_{k}^{2})$ on
$G,$ where the system of vector fields $X=\\{X_{i}\\}_{i=1}^{k}$ satisfies the
Hörmander condition of step $\kappa$. A symbol $\sigma$ belongs to the
subelliptic Hörmander class
${S}^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$ if it satisfies the
following symbol inequalities

$p_{\alpha,\beta,\rho,\delta,m,\textnormal{left}}(\sigma)^{\prime}:=\sup_{(x,[\xi])\in
G\times\widehat{G}}\|\widehat{\mathcal{M}}(\xi)^{(\rho|\alpha|-\delta|\beta|-m)}\partial_{X}^{(\beta)}\Delta_{\xi}^{\alpha}\sigma(x,\xi)\|_{\textnormal{op}}<\infty,\,\forall\alpha,\beta,$
(2.12)

and

$p_{\alpha,\beta,\rho,\delta,m,\textnormal{right}}(\sigma)^{\prime}:=\sup_{(x,[\xi])\in
G\times\widehat{G}}\|(\partial_{X}^{(\beta)}\Delta_{\xi}^{\alpha}\sigma(x,\xi))\widehat{\mathcal{M}}(\xi)^{(\rho|\alpha|-\delta|\beta|-m)}\|_{\textnormal{op}}<\infty,\,\forall\alpha,\beta.$
(2.13)

The family of seminorms

$p_{\alpha,\beta,\rho,\delta,m,\textnormal{right}}(\sigma)^{\prime},\,p_{\alpha,\beta,\rho,\delta,m,\textnormal{right}}(\sigma)^{\prime}$
(2.14)

endow the class ${S}^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$ with a
natural structure of Fréchet space. By denoting

$\Psi^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}):=\\{A:\sigma_{A}\in
S^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})\\},$ (2.15)

and by defining

$p_{\alpha,\beta,\rho,\delta,m,\textnormal{right}}(A)^{\prime}:=p_{\alpha,\beta,\rho,\delta,m,\textnormal{right}}(\sigma_{A})^{\prime},\quad
p_{\alpha,\beta,\rho,\delta,m,\textnormal{left}}(A)^{\prime}:=p_{\alpha,\beta,\rho,\delta,m,\textnormal{left}}(\sigma_{A})^{\prime},$
(2.16)

the operator class $\Psi^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$
also has a structure of Fréchet space.

In the next theorem we summarise some of the fundamental properties of the
subelliptic calculus in [CR20] that we will use in our further analysis.

######  Theorem 2.11.

Let $0\leqslant\delta<\rho\leqslant 1,$ and let us denote
$\Psi^{m,\mathcal{L}}_{\rho,\delta}:=\textnormal{Op}({S}^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})),$
for every $m\in\mathbb{R}.$ Then,

  * (i)

The mapping $A\mapsto
A^{*}:\Psi^{m,\mathcal{L}}_{\rho,\delta}\rightarrow\Psi^{m,\mathcal{L}}_{\rho,\delta}$
is a continuous linear mapping between Fréchet spaces and the symbol of
$A^{*},$ $\sigma_{A^{*}}(x,\xi)$ satisfies the asymptotic expansion,

$\widehat{A^{*}}(x,\xi)\sim\sum_{|\alpha|=0}^{\infty}\Delta_{\xi}^{\alpha}\partial_{X}^{(\alpha)}(\widehat{A}(x,\xi)^{*}).$

This means that, for every $N\in\mathbb{N},$ and all $\ell\in\mathbb{N},$

$\footnotesize{\Delta_{\xi}^{\alpha_{\ell}}\partial_{X}^{(\beta)}\left(\widehat{A^{*}}(x,\xi)-\sum_{|\alpha|\leqslant
N}\Delta_{\xi}^{\alpha}\partial_{X}^{(\alpha)}(\widehat{A}(x,\xi)^{*})\right)\in{S}^{m-(\rho-\delta)(N+1)-\rho\ell+\delta|\beta|,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})},$

where $|\alpha_{\ell}|=\ell.$

  * (ii)

The mapping $(A_{1},A_{2})\mapsto A_{1}\circ
A_{2}:\Psi^{m_{1},\mathcal{L}}_{\rho,\delta}\times\Psi^{m_{2},\mathcal{L}}_{\rho,\delta}\rightarrow\Psi^{m_{3},\mathcal{L}}_{\rho,\delta}$
is a continuous bilinear mapping between Fréchet spaces, and the symbol of
$A=A_{1}\circ A_{2},$ satisfies the asymptotic expansion,

$\sigma_{A}(x,\xi)\sim\sum_{|\alpha|=0}^{\infty}(\Delta_{\xi}^{\alpha}\widehat{A}_{1}(x,\xi))(\partial_{X}^{(\alpha)}\widehat{A}_{2}(x,\xi)),$

this means that, for every $N\in\mathbb{N},$ and all $\ell\in\mathbb{N},$

$\displaystyle\Delta_{\xi}^{\alpha_{\ell}}\partial_{X}^{(\beta)}\left(\sigma_{A}(x,\xi)-\sum_{|\alpha|\leqslant
N}(\Delta_{\xi}^{\alpha}\widehat{A}_{1}(x,\xi))(\partial_{X}^{(\alpha)}\widehat{A}_{2}(x,\xi))\right)$
$\displaystyle\hskip
56.9055pt\in{S}^{m_{1}+m_{2}-(\rho-\delta)(N+1)-\rho\ell+\delta|\beta|,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}),$

for every $\alpha_{\ell}\in\mathbb{N}_{0}^{n}$ with $|\alpha_{\ell}|=\ell.$

  * (iii)

For $0\leqslant\delta<\rho\leqslant 1,$ (or for $0\leq\delta\leq\rho\leq 1,$
$\delta<1/{\kappa}$) let us consider a continuous linear operator
$A:C^{\infty}(G)\rightarrow\mathscr{D}^{\prime}(G)$ with symbol
$\sigma\in{S}^{0,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$. Then $A$
extends to a bounded operator from $L^{2}(G)$ to $L^{2}(G).$

Now, we introduce the family of subelliptic pseudo-differential operators that
we will investigate in the setting of the Dixmier trace.

######  Definition 2.12.

Let $m\in\mathbb{R},$ and let $0\leqslant\delta<\rho\leqslant 1.$ Let
$a=a(x,\xi)\in{S}^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}).$ Assume
also that $a(x,\xi)$ is invertible for every $(x,[\xi])\in
G\times\widehat{G},$ and satisfies

$\sup_{(x,[\xi])\in
G\times\widehat{G}}\|\widehat{\mathcal{M}}(\xi)^{m}a(x,\xi)^{-1}\|_{\textnormal{op}}<\infty.$
(2.17)

Then we will say that the operator $A$ associated with $a(\cdot,\cdot)$ is
$\mathcal{L}$-elliptic.

######  Remark 2.13.

Let $A$ be an $\mathcal{L}$-elliptic operator. Then, Proposition 7.1 in [CR20,
Page 77] shows that there exists
$B\in{S}^{-m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}),$ such that
$AB-I,BA-I\in{S}^{-\infty,\mathcal{L}}(G\times\widehat{G}).$ Moreover, the
symbol of $B$ satisfies the following asymptotic expansion

$\widehat{B}(x,\xi)\sim\sum_{N=0}^{\infty}\widehat{B}_{N}(x,\xi),\,\,\,(x,[\xi])\in
G\times\widehat{G},$ (2.18)

where
$\widehat{B}_{N}(x,\xi)\in{S}^{-m-(\rho-\delta)N,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$
obeys to the recursive formula

$\widehat{B}_{N}(x,\xi)=-a(x,\xi)^{-1}\left(\sum_{k=0}^{N-1}\sum_{|\gamma|=N-k}(\Delta_{\xi}^{\gamma}a(x,\xi))(\partial_{X}^{(\gamma)}\widehat{B}_{k}(x,\xi))\right),\,\,N\geqslant
1,$ (2.19)

with $\widehat{B}_{0}(x,\xi)=a(x,\xi)^{-1}.$

###  2.3. Determinants on Hilbert spaces

Since we are interested in the study of the Dixmier trace and its regularised
determinant we will require some preliminary basic notions. First, we recall
the definition of Schatten-von Neumann ideals on Hilbert spaces. A reason for
this is that we will use operators $T$ of Dixmier type, where $T^{p}$ is of
class trace.

Let ${\mathcal{H}}$ be a complex separable Hilbert space endowed with an inner
product denoted by ${\left({\cdot,\cdot}\right)}$, and let
$T:{\mathcal{H}}\rightarrow{\mathcal{H}}$ be a linear compact operator. Let
$T^{*}:{\mathcal{H}}\rightarrow{\mathcal{H}}$ be the adjoint of $T$. Then the
linear operator $(T^{*}T)^{\frac{1}{2}}:{\mathcal{H}}\rightarrow{\mathcal{H}}$
is positive and compact. Let $(\psi_{k})_{k}$ be an orthonormal basis for
${\mathcal{H}}$ consisting of eigenvectors of $|T|=(T^{*}T)^{\frac{1}{2}}$,
and let $s_{k}(T)$ be the eigenvalue corresponding to the eigenvector
$\psi_{k}$, $k=1,2,\dots$. The non-negative numbers $s_{k}(T)$, $k=1,2,\dots$,
are called the singular values of $T:{\mathcal{H}}\rightarrow{\mathcal{H}}$.
If

$\sum_{k=1}^{\infty}s_{k}(T)<\infty,$

then the linear operator $T:{\mathcal{H}}\rightarrow{\mathcal{H}}$ is said to
be in the trace class $S_{1}(\mathcal{H})$. It can be shown that
$S_{1}({\mathcal{H}})$ is a Banach space in which the norm $\|\cdot\|_{S_{1}}$
is given by

$\|T\|_{S_{1}}=\sum_{k=1}^{\infty}s_{k}(T),\,T\in S_{1}(\mathcal{H}),$

with multiplicities counted. Let $T:{\mathcal{H}}\rightarrow{\mathcal{H}}$ be
an operator in $S_{1}({\mathcal{H}})$ and let $(\phi_{k})_{k}$ be any
orthonormal basis for ${\mathcal{H}}$. Then, the series
$\sum\limits_{k=1}^{\infty}{\left({T\phi_{k},\phi_{k}}\right)}$ is absolutely
convergent and the sum is independent of the choice of the orthonormal basis
$(\phi_{k})_{k}$. Thus, we can define the trace $\operatorname{Tr}(T)$ of any
linear operator $T:{\mathcal{H}}\rightarrow{\mathcal{H}}$ in
$S_{1}(\mathcal{H})$ by

$\operatorname{Tr}(T):=\sum_{k=1}^{\infty}{\left({T\phi_{k},\phi_{k}}\right)},$

where $\\{\phi_{k}:k=1,2,\dots\\}$ is any orthonormal basis for
${\mathcal{H}}$. If the singular values are square-summable $T$ is called a
Hilbert-Schmidt operator. It is clear that every trace class operator is a
Hilbert-Schmidt operator.

More generally, if $1\leq p<\infty$ and the sequence of singular values is
$p$-summable, then $T$ is said to belong to the Schatten-von Neumann class
${S}_{p}({\mathcal{H}})$. Indeed, it is well known that each
${S}_{p}({\mathcal{H}})$ is an ideal in $\mathcal{L}({\mathcal{H}})$. If
$1\leq p<\infty$, a norm is associated to ${S}_{p}({\mathcal{H}})$ by

$\|T\|_{S_{p}}=\left(\sum\limits_{k=1}^{\infty}(s_{k}(T))^{p}\right)^{\frac{1}{p}}.$

If $1\leq p<\infty$ the class $S_{p}({\mathcal{H}})$ becomes a Banach space
endowed by the norm $\|T\|_{S_{p}}$. If $p=\infty$ we define
$S_{\infty}({\mathcal{H}})$ as the class of bounded linear operators on $H$,
with $\|T\|_{S_{\infty}}:=\|T\|_{\textnormal{op}}$, the operator norm.

######  Remark 2.14.

We note that for a positive compact operator $A$ on $\mathcal{H}$ and $1\leq
p<\infty,$ $A^{p}\in S_{1}(\mathcal{H}),$ if and only $A\in
S_{p}(\mathcal{H}),$

For operators in the class ${S}_{1}(\mathcal{H})$ we have the following
determinant formula, see [GGK00, Page 19].

######  Theorem 2.15 (Plemelj-Smithies formula).

If $T\in S_{1}({\mathcal{H}})$ one has

$\det(I+\lambda
T)=\exp\left(\sum\limits_{m=1}^{\infty}\frac{(-1)^{m+1}}{m}\operatorname{Tr}(T^{m})\lambda^{m}\right),$

for $\lambda\in\mathbb{C}$ with $|\lambda|$ small enough.

###  2.4. Determinant of invariant operators on Hilbert spaces

In this section we present the determinant notion in the setting of invariant
operators on Hilbert spaces as introduced in [DR18]. We record the following
theorem:

######  Theorem 2.16.

Let ${\mathcal{H}}$ be a complex Hilbert space and let
${\mathcal{H}}^{\infty}\subset{\mathcal{H}}$ be a dense linear subspace of
${\mathcal{H}}$. Let
$\\{d_{j}\\}_{j\in{{\mathbb{N}}}_{0}}\subset{{\mathbb{N}}}$ and let
$\\{e_{j}^{k}\\}_{j\in{{\mathbb{N}}}_{0},1\leq k\leq d_{j}}$ be an orthonormal
basis of ${\mathcal{H}}$ such that $e_{j}^{k}\in{\mathcal{H}}^{\infty}$ for
all $j$ and $k$. Let $H_{j}:={\rm span}\\{e_{j}^{k}\\}_{k=1}^{d_{j}}$, and let
$P_{j}:{\mathcal{H}}\to H_{j}$ be the orthogonal projection. For
$f\in{\mathcal{H}}$, we denote
$\widehat{f}(j,k):=(f,e_{j}^{k})_{{\mathcal{H}}}$ and let
$\widehat{f}(j)\in\mathbb{C}^{d_{j}}$ denote the column of $\widehat{f}(j,k)$,
$1\leq k\leq d_{j}.$ Let $T:{\mathcal{H}}^{\infty}\to{\mathcal{H}}$ be a
linear operator. Then the following conditions are equivalent:

  * (A)

For each $j\in\mathbb{N}_{0}$, we have $T(H_{j})\subset H_{j}$.

  * (B)

For each $\ell\in\mathbb{N}_{0}$ there exists a matrix
$\sigma_{T}(\ell)\in\mathbb{C}^{d_{\ell}\times d_{\ell}}$ such that for all
$e_{j}^{k}$

$\widehat{Te_{j}^{k}}(\ell,m)=\sigma_{T}(\ell)_{mk}\delta_{j\ell}.$

  * (C)

For each $\ell\in\mathbb{N}_{0}$ there exists a matrix
$\sigma_{T}(\ell)\in\mathbb{C}^{d_{\ell}\times d_{\ell}}$ such that

$\widehat{Tf}(\ell)=\sigma_{T}(\ell)\widehat{f}(\ell)$

for all $f\in{\mathcal{H}}^{\infty}.$

The matrices $\sigma_{T}(\ell)$ in (B) and (C) coincide.

The equivalent properties (A)–(C) follow from the condition

  * (D)

For each $j\in\mathbb{N}_{0}$, we have $TP_{j}=P_{j}T$ on
${\mathcal{H}}^{\infty}$.

If, in addition, $T$ extends to a bounded operator
$T\in{\mathscr{L}}({\mathcal{H}})$ then (D) is equivalent to (A)–(C).

Under the assumptions of Theorem 2.16, we have the direct sum decomposition

${\mathcal{H}}=\bigoplus_{j=0}^{\infty}H_{j},\quad H_{j}={\rm
span}\\{e_{j}^{k}\\}_{k=1}^{d_{j}},$ (2.20)

and we have $d_{j}=\dim H_{j}.$ The two main applications that we will
consider correspond to ${\mathcal{H}}=L^{2}(M)$ for a compact manifold $M$
with $H_{j}$ being the eigenspaces of an elliptic pseudo-differential operator
$E$, or with ${\mathcal{H}}=L^{2}(G)$ for a compact Lie group $G$ with
$H_{j}=\textrm{span}\\{\xi_{km}\\}_{1\leq k,m\leq d_{\xi}}$ for a unitary
irreducible representation $\xi\in[\xi_{j}]\in\widehat{G}$. The difference is
that in the first case we will have that the eigenvalues of $E$ corresponding
to $H_{j}$’s are all distinct, while in the second case the eigenvalues of the
Laplacian on $G$ for which $H_{j}$’s are the eigenspaces, may coincide.

######  Definition 2.17.

An operator $T$ satisfying any of the equivalent properties (A)–(C) in Theorem
2.16, will be called an invariant operator, or a Fourier multiplier relative
to the decomposition $\\{H_{j}\\}_{j\in{{\mathbb{N}}}_{0}}$ in (2.20). If the
collection $\\{H_{j}\\}_{j\in{{\mathbb{N}}}_{0}}$ is fixed once and for all,
we can just say that $T$ is invariant or a Fourier multiplier.

The family of matrices $\sigma$ will be called the matrix symbol of $T$
relative to the partition $\\{H_{j}\\}$ and to the basis $\\{e_{j}^{k}\\}$. It
is an element of the space $\Sigma$ defined by

$\Sigma=\\{\sigma:{{\mathbb{N}}}_{0}\ni\ell\mapsto\sigma(\ell)\in\mathbb{C}^{d_{\ell}\times
d_{\ell}}\\}.$ (2.21)

In this section we will investigate the concept of determinant on embedded in
the sense of [GGK00] for the notion of invariant operators introduced in
[DR18].

In view of Theorem 2.16 we have the following determinant formula for
$E$-invariant operators that later on will be used to establish a determinant
formula in terms of the Dixmier trace (for the proof, see [CDR21]).

######  Theorem 2.18.

If $T\in S_{1}({\mathcal{H}})$ is invariant, then

$\det(I+\lambda
T)=\exp\left(\sum\limits_{m=1}^{\infty}\frac{(-1)^{m+1}}{m}\lambda^{m}\sum\limits_{\ell=0}^{\infty}\operatorname{Tr}(\sigma_{T}(\ell)^{m})\right),$

for $\lambda\in\mathbb{C}$ with $|\lambda|$ small enough.

###  2.5. The Dixmier ideal on Hilbert spaces

By following Connes [Con94], if $\mathcal{H}$ is a Hilbert space (we are
interested in $\mathcal{H}=L^{2}(M)$ where $M$ is a closed manifold of
dimension $n$), the class $\mathcal{L}^{(1,\infty)}(\mathcal{H})$ consists of
those compact linear operators $A$ on $\mathcal{H}$ satisfying

$\sum_{1\leq n\leq N}s_{n}(A)=O(\log(N)),\,\,\,N\rightarrow\infty,$ (2.22)

where $\\{s_{n}(A)\\}$ denotes the sequence of singular values of $A$, i.e.
the square roots of the eigenvalues of the positive-definite self-adjoint
operator $A^{\ast}A.$ So, $\mathcal{L}^{(1,\infty)}(\mathcal{H})$ is endowed
with the norm

$\|A\|_{\mathcal{L}^{(1,\infty)}(\mathcal{H})}=\sup_{N\geq
2}\frac{1}{\log(N)}\sum_{1\leq n\leq N}s_{n}(A).$ (2.23)

For an operator $A\in\mathcal{L}^{(1,\infty)}(\mathcal{H})$ the sequence

$\alpha_{N}(A):=\frac{1}{\log(N)}\sum_{j=1}^{N}s_{j}(A),\quad N\in\mathbb{N},$
(2.24)

is thus bounded. If the limit of the sequence $\\{\alpha_{N}(A)\\}$ when
$N\rightarrow\infty,$ does exist we will define

$\displaystyle\textnormal{\bf
Tr}_{\textnormal{Dix}}(A):=\lim_{N\rightarrow\infty}\frac{1}{\log(N)}\sum_{j=1}^{N}s_{j}(A).$

We recall the main result in Comptes Rendus’s note of Dixmier [Dix66]
regarding the existence of a trace functional different from the spectral
trace defined on the ideal $\mathcal{L}^{(1,\infty)}(\mathcal{H})$ which
contains the ideal of trace class operators $S_{1}(\mathcal{H})$.

######  Theorem 2.19 (Dixmier [Dix66]).

Let $\omega\in\ell^{\infty}(\mathbb{N}\setminus\\{0\\})^{*}$ be a bounded
functional satisfying the following conditions:

  * (D1):

$\omega$ is a positive linear functional that satisfies
$\omega(1,1,\cdots,1,\cdots)=1.$

  * (D2):

$\omega((a_{n})_{n\in\mathbb{N}\setminus\\{0\\}})=0,$ for a sequence
$(a_{n})_{n\in\mathbb{N}\setminus\\{0\\}}\in\ell^{\infty}(\mathbb{N}\setminus\\{0\\})$
if

$\lim_{N\rightarrow\infty}a_{n}=0.$

  * (D3):

For any
$(a_{n})_{n\in\mathbb{N}\setminus\\{0\\}}\in\ell^{\infty}(\mathbb{N}\setminus\\{0\\}),$

$\omega(a_{1},a_{2},\cdots,a_{n},\cdots)=\omega(a_{1},a_{1},a_{2},a_{2},\cdots,a_{n},a_{n},\cdots).$

For a compact positive operator $A$ in $\mathcal{L}^{(1,\infty)}(\mathcal{H})$
we set

$\textnormal{\bf
Tr}_{\omega}(A):=\omega\left(\left\\{\frac{1}{\log(N)}\sum_{j=1}^{N}s_{j}(A)\right\\}_{N\geq
1}\right).$ (2.25)

Then, $\textnormal{\bf Tr}_{\omega}$ extends by linearity to a trace on
$\mathcal{L}^{(1,\infty)}(\mathcal{H}),$ and

$\textnormal{\bf Tr}_{\omega}(A)=\textnormal{\bf
Tr}_{\textnormal{Dix}}(A):=\lim_{N\rightarrow\infty}\frac{1}{\log(N)}\sum_{j=1}^{N}s_{j}(A),$
(2.26)

provided that the limit in the right hand side exists. In this case, the value
of $\textnormal{\bf Tr}_{\omega}(A)$ is independent of the choice of $\omega.$
Moreover, $\textnormal{\bf Tr}_{\omega}(A)=0,$ if $A\in S_{1}(\mathcal{H}).$

######  Remark 2.20.

Let us describe the extension of $\textnormal{\bf Tr}_{\omega}$ from the
positive elements in $\mathcal{L}^{(1,\infty)}(\mathcal{H})$ to general
compact operators in this ideal, see [Con94]. We use the decomposition of $A$
into its real and imaginary part,

$\textnormal{Re}(A):=\frac{A+A^{*}}{2},\,\,\textnormal{Im}(A):=\frac{A-A^{*}}{2i},$

and the decomposition of $\textnormal{Re}(A)$ and $\textnormal{Im}(A)$ into
their positive and negative parts,

$\displaystyle\textnormal{Re}(A)^{+}:=\frac{\textnormal{Re}(A)+|\textnormal{Re}(A)|}{2},\,\,\textnormal{Re}(A)^{-}:=\frac{|\textnormal{Re}(A)|-\textnormal{Re}(A)}{2},$

and

$\displaystyle\textnormal{Im}(A)^{+}:=\frac{\textnormal{Im}(A)+|\textnormal{Im}(A)|}{2},\,\,\textnormal{Im}(A)^{-}:=\frac{|\textnormal{Im}(A)|-\textnormal{Im}(A)}{2}.$

Now, the operator $A$ can be written as

$\displaystyle A$ $\displaystyle=\textnormal{Re}(A)+i\textnormal{Im}(A)$
$\displaystyle=\left(\textnormal{Re}(A)^{+}-\textnormal{Re}(A)^{-}\right)+i\left(\textnormal{Im}(A)^{+}-\textnormal{Im}(A)^{-}\right).$

So, by the linearity of the Dixmier trace $\textnormal{Tr}_{\omega}$ we have

$\displaystyle\textnormal{Tr}_{\omega}(A)$
$\displaystyle=\textnormal{Tr}_{\omega}(\textnormal{Re}(A))+i\textnormal{Tr}_{\omega}(\textnormal{Im}(A))$
$\displaystyle=\left(\textnormal{Tr}_{\omega}(\textnormal{Re}(A)^{+})-\textnormal{Tr}_{\omega}(\textnormal{Re}(A)^{-})\right)+i\left(\textnormal{Tr}_{\omega}(\textnormal{Im}(A)^{+})-\textnormal{Tr}_{\omega}(\textnormal{Im}(A)^{-})\right).$

######  Definition 2.21.

We denote by

$\mathcal{L}^{(1,\infty)}_{\omega}(\mathcal{H}):=\\{A\in\mathcal{L}^{(1,\infty)}(\mathcal{H}):\textnormal{\bf{Tr}}_{\textnormal{Dix}}(A)<\infty\\}.$
(2.27)

the sub-class of operators in $A\in\mathcal{L}^{(1,\infty)}(\mathcal{H}),$
such that the limit in the right-hand side of (2.26) exists, and in view of
Dixmier’s Theorem 2.19, we have

$\textnormal{\bf
Tr}_{\omega}(A)=\lim_{N\rightarrow\infty}\frac{1}{\log(N)}\sum_{j=1}^{N}s_{j}(A),$

for all $A\in\mathcal{L}^{(1,\infty)}_{\omega}(\mathcal{H}).$

######  Remark 2.22.

A remarkable connection between the Dixmier functional and the usual trace on
the ideal $S_{1}(\mathcal{H})$ is the following formula due to Connes and
Moscovici

$\textnormal{\bf{Tr}}_{\omega}(A)=\lim_{p\rightarrow
1^{+}}(p-1)\textnormal{\bf{Tr}}(A^{p}),\,\,A\geq 0,$ (2.28)

provided that $A^{p}\in S_{1}(\mathcal{H}),$ for all $1<p<\infty,$ (cf. [SU16,
Page 114]). Indeed, in view of Proposition 4 of [Con94, Page 313], for $A\geq
0,$ the existence of the limit in the right-hand side of (2.28) is equivalent
to the fact that $A\in\mathcal{L}^{(1,\infty)}_{\omega}(H).$

We finish this subsection by introducing the regularised determinant for the
Dixmier trace, which can be obtained from the Plemelj-Smithies formula in
Theorem 2.15. First, we prove the following lemma.

######  Lemma 2.23.

Let $T$ be a positive compact operator on a Hilbert space $H,$ and let us
assume that $T\in\mathcal{L}^{(1,\infty)}(\mathcal{H}).$ If $T^{p}\in
S_{1}(\mathcal{H}),$ for all $1<p<\infty$, then the function

$\textnormal{Det}_{p,\omega,T}(\lambda):=(\textnormal{Det}(1+\lambda
T^{p}))^{p-1},$ (2.29)

is analytic for $\lambda\in\mathbb{C}$ with $|\lambda|$ small enough.
Moreover, we have

$\textnormal{Det}_{\omega,T}(\lambda):=\lim_{p\rightarrow
1^{+}}\textnormal{Det}_{p,\omega,T}(\lambda)=\exp(\textnormal{\bf{Tr}}_{\omega}(T)\lambda),$
(2.30)

for $|\lambda|$ small enough. In particular,

$\textnormal{Det}_{\omega,T}^{\prime}(0):=\frac{d}{d\lambda}\textnormal{Det}_{\omega,T}(\lambda)|_{\lambda=0}=\textnormal{\bf{Tr}}_{\omega}(T).$

###### Proof.

Observe that in view of Theorem 2.19,
$\textnormal{\bf{Tr}}_{\omega}(T^{p})=0,$ for all $1<p<\infty$, since in this
case $T^{p}\in S_{1}(\mathcal{H}).$ In view of Theorem 2.15, the function
$\textnormal{Det}(1+\lambda T^{p})$ is analytic for $\lambda\in\mathbb{C}$
with $|\lambda|$ small enough. So, $\textnormal{Det}_{p,\omega,T}(\lambda)$
also is analytic in a small disc centered at zero and one has

$(\det(I+\lambda
T^{p}))^{p-1}=\left(\exp\left(\sum\limits_{m=1}^{\infty}\frac{(-1)^{m+1}}{m}\operatorname{Tr}(T^{pm})\lambda^{m}\right)\right)^{p-1},$

for $\lambda\in\mathbb{C}$ with $|\lambda|$ small enough. Because of

$\displaystyle(\det(I+\lambda T^{p}))^{p-1}$
$\displaystyle=\exp\left({(p-1)}\left(\sum\limits_{m=1}^{\infty}\frac{(-1)^{m+1}}{m}\operatorname{Tr}(T^{pm})\lambda^{m}\right)\right)$
$\displaystyle=\exp\left(\sum\limits_{m=1}^{\infty}\frac{(-1)^{m+1}}{m}(p-1)\operatorname{Tr}(T^{pm})\lambda^{m}\right),$

and by using (2.28) we see that

$\displaystyle\lim_{p\rightarrow 1^{+}}(\det(I+\lambda T^{p}))^{p-1}$
$\displaystyle=\exp\left(\sum\limits_{m=1}^{\infty}\frac{(-1)^{m+1}}{m}\lim_{p\rightarrow
1^{+}}(p-1)\operatorname{Tr}(T^{pm})\lambda^{m}\right)$
$\displaystyle=\exp\left(\sum\limits_{m=1}^{\infty}\frac{(-1)^{m+1}}{m}\textnormal{\bf{Tr}}_{\omega}(T^{m})\lambda^{m}\right).$

The assumption that $T^{p}\in S_{1}(\mathcal{H}),$ for all $1<p<\infty$,
implies that $\textnormal{\bf{Tr}}_{\omega}(T^{m})=0,$ for all $m>1,$ and that

$\textnormal{Det}_{\omega,T}(\lambda)=\exp\left(\sum\limits_{m=1}^{\infty}\frac{(-1)^{m+1}}{m}\textnormal{\bf{Tr}}_{\omega}(T^{m})\lambda^{m}\right)=\exp(\lambda\textnormal{\bf{Tr}}_{\omega}(T)).$

Thus, we end the proof. ∎

In view of Lemma 2.23 let us introduce the following definition.

######  Definition 2.24.

Let $\mathcal{L}^{(1,\infty)}_{adm}(\mathcal{H})$ be the family of the
positive compact operators $T$ on $H,$ such that
$T\in\mathcal{L}^{(1,\infty)}(\mathcal{H}),$ and $T^{p}\in
S_{1}(\mathcal{H}),$ for all $1<p<\infty$. In view of (2.30), the function

$\textnormal{Det}_{\omega,T}(\lambda):=\lim_{p\rightarrow
1^{+}}\textnormal{Det}_{p,\omega,T}(\lambda)=\exp(\textnormal{\bf{Tr}}_{\omega}(T)\lambda),$
(2.31)

admits analytic extension to the complex plane, which is given by

$\textnormal{Det}_{\omega,T}(z):=\exp(\textnormal{\bf{Tr}}_{\omega}(T)z),\,z\in\mathbb{C}.$
(2.32)

Define the regularised determinant $\textnormal{\bf Det}_{\omega}$ on the
family
$I+\mathcal{L}^{(1,\infty)}_{adm}(\mathcal{H})=\\{I+T:T\in\mathcal{L}^{(1,\infty)}_{adm}(\mathcal{H})\\}$
by

$\textnormal{\bf Det}_{\omega}(I+T):=\textnormal{Det}_{\omega,T}(1).$ (2.33)

######  Remark 2.25.

We can think of the functional $\textnormal{\bf Det}_{\omega}$ as a
regularised determinant for the Dixmier trace $\textnormal{\bf Tr}_{\omega}$.

###  2.6. Connes equivalence theorem

A remarkable result due to Connes computes the Dixmier class for the classical
pseudo-differential operators on a closed manifold.

We record that for a compact orientable manifold without boundary $M$ of
dimension $n,$ a pseudo-differential operator $A$ on $M$ can be defined by
using the notion of a local symbol, this means that for any local chart $U$,
the operator $A$ has the form

$Au(x)=\int\limits_{T^{*}_{x}U}e^{2\pi
ix\cdot\xi}\sigma^{A}(x,\xi)\widehat{u}(\xi)\,d\xi.$

The pseudo-differential operator $A$ is called classical, if $\sigma^{A}$
admits an asymptotic expansion
$\sigma^{A}(x,\xi)\sim\sum_{j=0}^{\infty}\sigma^{A}_{m-j}(x,\xi)$ in such a
way that each function $\sigma_{m-j}(x,\xi)$ is homogeneous in $\xi$ of order
$m-j$ for $\xi\neq 0$. The set of classical pseudo-differential operators of
order $m$ is denoted by $\Psi^{m}_{cl}(M)$, and we denote by
$\Psi^{m}_{+e}(M)$ the class of positive elliptic classical pseudo-
differential operators of order $m\in\mathbb{R}.$ For $A\in\Psi_{cl}^{m}(M)$,
and for $x\in M$, $\int_{|\xi|=1}\sigma_{-n}(x,\xi)\,d\xi$ defines a local
density which can be glued over $M$. In this case, the non-commutative residue
of $A$ is defined by the expression

$\textnormal{res}\,(A)=\frac{1}{n(2\pi)^{n}}\int\limits_{M}\int\limits_{|\xi|=1}\sigma^{A}_{-n}(x,\xi)\,d\xi\,dx.$
(2.34)

Classical pseudo-differential operators with order $-n,$ where $n=\dim(M)$ are
in the Dixmier class $\mathcal{L}^{(1,\infty)}(L^{2}(M)),$ and moreover:

######  Theorem 2.26 (Connes, [Con94]).

If $A\in\Psi^{-n}_{cl}(M)$ is a classical pseudo-differential operator, then

$\textnormal{res}(A)=\textnormal{\bf{Tr}}_{\omega}(A),$ (2.35)

for any functional $\omega$ satisfying the conditions in Dixmier’s Theorem
2.19.

##  3\. Dixmier traces and Wodzicki residues on compact Lie groups

In this section we investigate the Dixmier traceability of pseudo-differential
operators on compact Lie groups. We will apply our analysis to the case of
subelliptic pseudo-differential classes
$\Psi^{-Q,\mathcal{L}}_{\rho,0}(G\times\widehat{G}).$ In particular, if we
replace the sub-Laplacian $\mathcal{L}$ by the Laplace-Beltrami operator our
results can be applied to the standard Hörmander classes
$\Psi^{-n}_{\rho,0}(G;\textnormal{loc})=\Psi^{-n}_{\rho,0}(G\times\widehat{G}).$

For our further analysis we require the following lemma (see Lemma 8.10 in
[CR20]).

######  Lemma 3.1.

For $0\leqslant\delta<\rho\leqslant 1,$ let us consider a continuous linear
operator $P:C^{\infty}(G)\rightarrow\mathscr{D}^{\prime}(G)$ with global
symbol $\sigma_{P}\in{S}^{m,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$,
of order $m<-Q.$ Then $\textnormal{\bf{Tr}}_{w}(P)=0.$

######  Remark 3.2 (Dixmier traces of Fourier multipliers).

Let $G$ be a compact Lie group and let $A:C^{\infty}(G)\rightarrow
C^{\infty}(G)$ be a Fourier multiplier. In terms of the quantisation process
in [RT10], there is a matrix-symbol
$\sigma_{A}:\widehat{G}\rightarrow\cup_{[\xi]\in\widehat{G}}\mathbb{C}^{d_{\xi}\times
d_{\xi}}$ such that

$Af(x)=\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{\bf{Tr}}[\xi(x)\sigma_{A}(\xi)\widehat{f}(\xi)],\,\,\,f\in
C^{\infty}(G).$ (3.1)

It was proved in [CDC20] that
$A\in\mathcal{L}^{(1,\infty)}_{\omega}(L^{2}(G)),$ (see Definition 2.21) if
and only if,

$\|\sigma_{A}(\xi)\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}:=\lim_{N\rightarrow\infty}\frac{1}{\log
N}\sum_{[\xi]:\langle\xi\rangle\leq
N}d_{\xi}\textnormal{\bf{Tr}}(|\sigma_{A}(\xi)|)<\infty,$ (3.2)

and that
$\textnormal{\bf{Tr}}_{\omega}(A)=\|\sigma_{A}(\xi)\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}.$
In view of Theorem 4.3, for $A\in\Psi^{-n}_{+e}(G),$ $n=\dim(G),$ we have

$\textnormal{Det}_{\omega,A}(\lambda)=\exp\left(\lambda\|\sigma_{A}(\xi)\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}\right),$
(3.3)

for $\lambda\in\mathbb{C}$ with $|\lambda|$ small enough. In particular,

$\textnormal{Det}_{\omega,A}^{\prime}(0):=\frac{d}{d\lambda}\textnormal{Det}_{\omega,A}(\lambda)|_{\lambda=0}=\|\sigma_{A}(\xi)\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}.$

Now, we will compute the Dixmier trace of positive operators.

######  Lemma 3.3.

Let $A\in\Psi^{-Q,\mathcal{L}}_{\rho,0}(G\times\widehat{G})$ be a positive
$\mathcal{L}$-elliptic pseudo-differential operator and let $0<\rho\leq 1$.
Assume that the symbol of $A$ admits an asymptotic expansion

$\sigma_{A}(x,[\xi])\sim\sum_{k=-\infty}^{-Q}\sigma_{k}(x,[\xi]),\,$

in components with decreasing order, which means that, for any
$N\in\mathbb{N},$

$\sigma_{A}(x,[\xi])-\sum_{k=-N-Q}^{-Q}\sigma_{k}(x,[\xi])\in
S^{-(N+1)\rho-Q}_{\rho,0}(G\times\widehat{G}),$ (3.4)

with the principal symbol $\sigma_{-Q}(x,[\xi])\geq 0,$ being positive. Then
$A\in\mathcal{L}^{(1,\infty)}(L^{2}(G)),$

$\textnormal{\bf{Tr}}_{\omega}(A)=\int\limits_{G}\|\sigma_{-Q}(x,[\xi])\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}dx,$
(3.5)

and we have

$\textnormal{Det}_{\omega,A}(\lambda)=\exp\left(\textnormal{\bf{Tr}}_{\omega}(A)\lambda\right)=\exp\left(\int\limits_{G}\|\sigma_{-Q}(x,[\xi])\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}dx\cdot\lambda\right),$
(3.6)

for $\lambda\in\mathbb{C}$ with $|\lambda|$ small enough, and

$\textnormal{Det}_{\omega,A}^{\prime}(0):=\frac{d}{d\lambda}\textnormal{Det}_{\omega,A}(\lambda)|_{\lambda=0}=\textnormal{\bf{Tr}}_{\omega}(A).$

###### Proof.

For any $x\in G,$ observe that

$\|\sigma_{-Q}(x,[\xi])\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}:=\lim_{N\rightarrow\infty}\frac{1}{\log
N}\sum_{[\xi]:\langle\xi\rangle\leq
N}d_{\xi}\textnormal{\bf{Tr}}(|\sigma_{-Q}(x,[\xi])|).$

The estimate

$\textnormal{\bf{Tr}}(|\sigma_{-Q}(x,[\xi])|\widehat{\mathcal{M}}(\xi)^{Q}\widehat{\mathcal{M}}(\xi)^{-Q})\leq\||\sigma_{-Q}(x,[\xi])|\widehat{\mathcal{M}}(\xi)^{Q}\|_{\textnormal{op}}\textnormal{\bf{Tr}}(|\sigma_{-Q}(x,[\xi])|)$

implies that

$\displaystyle\|\sigma_{-Q}(x,[\xi])\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}$
$\displaystyle=\||\sigma_{-Q}(x,[\xi])|\widehat{\mathcal{M}}(\xi)^{Q}\widehat{\mathcal{M}}(\xi)^{-Q}\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}$
$\displaystyle\leq\left(\sup_{[\xi]\in\widehat{G}}\|\,|\sigma_{-Q}(x,[\xi])|\,\widehat{\mathcal{M}}(\xi)^{Q}\|_{\textnormal{op}}\right)\|\widehat{\mathcal{M}}(\xi)^{-Q}\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}$
$\displaystyle\lesssim 1.$

Indeed, the functional calculus gives us that $|\sigma_{-Q}(x,[\xi])|\in
S^{-Q}_{\rho,0}(G\times\widehat{G}),$ and then, that

$\sup_{[\xi]\in\widehat{G}}\|\,|\sigma_{-Q}(x,[\xi])|\,\widehat{\mathcal{M}}(\xi)^{Q}\|_{\textnormal{op}}\leq
C<\infty,$

uniformly in $x\in G.$ While, the estimate
$\|\widehat{\mathcal{M}}(\cdot)^{-Q}\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}<\infty$
was proved in Lemma 8.9 of [CR20]. Consequently, the compactness of the group
$G,$ implies that
$\|\sigma_{-Q}(x,[\xi])\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}\in
L^{1}(G).$ So, by the positivity of $A,$ if we prove (3.5), we deduce that
$A\in\mathcal{L}^{(1,\infty)}_{\omega}(L^{2}(G))$ in view of Remark 2.22.

In view of the asymptotic expansion (3.4), we can decompose the operator $A$
into the sum $A=A_{0}+P,$ where $A_{0}:=\textnormal{Op}(\sigma_{-Q}),$ and $P$
a pseudo-differential operator in the class
$\Psi^{-Q-\rho,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}).$ Before
continuing with the proof, let us remark that in view of the linearity of the
functional $\textnormal{\bf{Tr}}_{\omega},$ we have

$\textnormal{\bf{Tr}}_{\omega}(A)=\textnormal{\bf{Tr}}_{\omega}(A_{0})+\textnormal{\bf{Tr}}_{\omega}(P)=\textnormal{\bf{Tr}}_{\omega}(A_{0}).$
(3.7)

Note that we have used that $\textnormal{\bf{Tr}}_{\omega}(P)=0$ in view of
Lemma 3.1. So, (3.7) is the reason of why the symbol $\sigma_{-Q}$ appears in
(3.6). Let is compute the Dixmier trace of $A.$

For every $z\in G,$ let us consider the Fourier multiplier $A_{z}$ associated
to the symbol $\sigma(z,\cdot):=\sigma_{-Q}(z,\cdot).$ Observe that any
Fourier multiplier $A_{z}$ is positive because $\sigma(z,[\xi])\geqslant 0,$
for every $[\xi].$ Also, the $\mathcal{L}$-ellipticity of $A$ implies the
$\mathcal{L}$-ellipticity of $A_{z}$ for every $z\in G.$ Because $A_{z}$ is
left-invariant and of subelliptic order $-Q,$ for all $z\in G,$ from Lemma 8.9
of [CR20] we have that $A_{z}\in\mathcal{L}^{(1,\infty)}(L^{2}(G)),$ and from
Theorem 1.1 of [CDC20] (see also Remark 3.2), we have that the Dixmier trace
of any $A_{z}\geq 0,$ is given by

$\textnormal{\bf{Tr}}_{\omega}(A_{z})=\lim_{p\rightarrow
1^{+}}(p-1)\textnormal{\bf{Tr}}(A_{z}^{p})=\|\sigma_{-Q}(z,\cdot)\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}.$
(3.8)

In view of the asymptotic expansion (3.4), the functional calculus in [CR20]
implies that

$A^{p}=\textnormal{Op}[(x,[\xi])\mapsto\sigma(x,[\xi])^{p}]+R_{p},\quad
A_{z}^{p}=\textnormal{Op}[[\xi]\mapsto\sigma_{-Q}(z,\xi)^{p}],$ (3.9)

where, for any $1<p<\infty,$ $R_{p}$ is a subelliptic pseudo-differential
operator of order $-Qp-\rho.$ Moreover, since the order of $A^{p}$ is $-Qp,$
for all $1<p<\infty,$ in view of Corollary 8.14 of [CR20], $A^{p}$ is trace
class and
$\textnormal{\bf{Tr}}(A^{p})=\int\limits_{G}\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{Tr}[\sigma_{A^{p}}(x,[\xi])]dx,$
in view of the trace formula in [DR14a]. Because $A\geq 0$ we have that

$\displaystyle\textnormal{\bf{Tr}}_{\omega}(A)$
$\displaystyle=\lim_{p\rightarrow
1^{+}}(p-1)\textnormal{\bf{Tr}}(A^{p})=\lim_{p\rightarrow
1^{+}}(p-1)\int\limits_{G}\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{Tr}[\sigma_{A^{p}}(x,[\xi])]dx$
$\displaystyle=\lim_{p\rightarrow
1^{+}}(p-1)\int\limits_{G}\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{Tr}[\sigma_{-Q}(x,[\xi])^{p}+\sigma_{R_{p}}(x,[\xi])]dx$
$\displaystyle=\lim_{p\rightarrow
1^{+}}(p-1)\left(\int\limits_{G}\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{Tr}[\sigma_{-Q}(x,[\xi])^{p}]dx+\int\limits_{G}\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{Tr}[\sigma_{R_{p}}(x,[\xi])]dx\right)$
$\displaystyle=\lim_{p\rightarrow
1^{+}}(p-1)\left(\textnormal{\bf{Tr}}[\textnormal{Op}(\sigma_{-Q}^{p})]+\textnormal{\bf{Tr}}[R_{p}]\right).$

Now, we are going to guarantee the existence of this limit. Firstly, using
again Corollary 8.14 of [CR20], and that the order of
$\textnormal{Op}(\sigma_{-Q}^{p})$ is $-Qp,$ for all $1<p<\infty,$ we have
that
$\textnormal{\bf{Tr}}(\textnormal{Op}(\sigma_{-Q}^{p}))=\int\limits_{G}\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{Tr}[\sigma_{-Q}^{p}(x,[\xi])]dx,$
in view of the trace formula in [DR14a]. Moreover, using (3.8) we have that

$\displaystyle\lim_{p\rightarrow
1^{+}}(p-1)\textnormal{\bf{Tr}}[\textnormal{Op}(\sigma_{-Q}^{p})]$
$\displaystyle=\lim_{p\rightarrow
1^{+}}(p-1)\int\limits_{G}\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{Tr}[\sigma_{-Q}(x,[\xi])^{p}]dx$
$\displaystyle=\int\limits_{G}\lim_{p\rightarrow
1^{+}}(p-1)\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{Tr}[\sigma_{-Q}(x,[\xi])^{p}]dx$
$\displaystyle=\int\limits_{G}\lim_{p\rightarrow
1^{+}}(p-1)\textnormal{\bf{Tr}}(A_{x}^{p})dx=\int\limits_{G}\textnormal{\bf{Tr}}_{\omega}(A_{x})dx$
$\displaystyle=\int\limits_{G}\|\sigma_{-Q}(x,\cdot)\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}dx.$

We observe that we can interchange the limit with the integral in the previous
argument in view of the dominated convergence theorem. Indeed, the operator
$\textnormal{Op}(\sigma_{-Q}(x,[\xi])^{p})$ of subelliptic order $m=-Qp,$ has
a right-convolution kernel $k_{x,p}$ that behaves like (view [CR20, Section
4])

$|k_{x,p}(y)|\lesssim_{\sigma_{A_{0}}}(1+|y|)^{-\frac{Q+m}{\rho}}=(1+|y|)^{\frac{(p-1)Q}{\rho}}\lesssim
1,$

for all $p\in(1,2).$ So, $k_{x,p}$ is locally integrable at the identity
$e_{G}$ for all $p\in(1,2),$ in view of the compactness of $G$ and the
continuity of the geodesic norm $y\mapsto|y|.$ Note that the Fourier inversion
formula gives

$k_{x,p}(e_{G})=\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{Tr}[\sigma_{-Q}(x,[\xi])^{p}].$
(3.10)

So, we have that $|k_{x,p}(e_{G})|\leq C,$ with $C>0,$ independent of
$p\in(1,2),$ and that $\int_{G}|k_{x,p}(e_{G})|dx\leq C$ for all $1<p\leq 2,$
in view of the compactness of $G.$ The dominated convergence theorem gives

$\lim_{p\rightarrow
1^{+}}(p-1)\int\limits_{G}k_{x,p}(e_{G})dx=\int\limits_{G}\lim_{p\rightarrow
1^{+}}(p-1)k_{x,p}(e_{G})dx$ (3.11)

ensuring that

$\lim_{p\rightarrow
1^{+}}(p-1)\int\limits_{G}\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{Tr}[\sigma_{-Q}(x,[\xi])^{p}]dx=\int\limits_{G}\lim_{p\rightarrow
1^{+}}(p-1)\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{Tr}[\sigma_{-Q}(x,[\xi])^{p}]dx$

and the identity

$\lim_{p\rightarrow
1^{+}}(p-1)\textnormal{\bf{Tr}}[\textnormal{Op}(\sigma_{-Q}^{p})]=\int\limits_{G}\|\sigma_{-Q}(x,\cdot)\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}dx.$
(3.12)

We claim that

$\lim_{p\rightarrow 1^{+}}(p-1)\textnormal{\bf{Tr}}(R_{p})=0.$

For the proof, we fix $p\in(1,2),$ let $0<\varepsilon<\frac{\rho}{2}$ and
observe that

$\displaystyle|\textnormal{\bf{Tr}}(R_{p})|=|\textnormal{\bf{Tr}}(R_{p}{\mathcal{M}}_{(Q+\varepsilon)p}{\mathcal{M}}_{-(Q+\varepsilon)p})|\leq
C\|R_{p}{\mathcal{M}}_{(Q+\varepsilon)p}\|_{\mathscr{B}(L^{2}(G))}|\textnormal{\bf{Tr}}({\mathcal{M}}_{-(Q+\varepsilon)p})|.$

Since

$R_{p}{\mathcal{M}}_{(Q+\varepsilon)p}\in\Psi^{-\rho+\varepsilon
p,\mathcal{L}}_{\rho,0}(G\times\widehat{G}),$ (3.13)

the inequalities $\varepsilon<p\varepsilon<2\varepsilon<\rho,$ and the
Calderón-vaillancourt theorem (see (iii) in Theorem 2.11) imply that

$\forall
1<p<2,\,\,\|R_{p}{\mathcal{M}}_{(Q+\varepsilon)p}\|_{\mathscr{B}(L^{2}(G))}<\infty.$

For any $x\in G,$ the Sobolev embedding theorem implies that

$\displaystyle|\textnormal{Op}[(x,[\xi])\mapsto$
$\displaystyle\sigma_{R_{p}}(x,[\xi])\widehat{\mathcal{M}}_{(Q+\varepsilon)p}(\xi)]f(x)|$
$\displaystyle\leq\sup_{z\in G}|\textnormal{Op}[[\xi]\mapsto$
$\displaystyle\sigma_{R_{p}}(z,[\xi])\widehat{\mathcal{M}}_{(Q+\varepsilon)p}(\xi)]f(x)|$
$\displaystyle\leq\sum_{|\alpha|\leq{[n/2]+1}}\|\textnormal{Op}[[\xi]\mapsto
X_{z}^{\alpha}\sigma_{R_{p}}(z,[\xi])\widehat{\mathcal{M}}_{(Q+\varepsilon)p}(\xi)]f(x)\|_{L^{2}(G_{z})}.$

The notation $L^{2}(G_{z})$ indicates that the $L^{2}$-norm is taken in the
$z$-variables. Consequently,

$\displaystyle\|R_{p}{\mathcal{M}}_{(Q+\varepsilon)p}\|_{\mathscr{B}(L^{2})}$
$\displaystyle=\sup_{\|f\|_{L^{2}}=1}\|\textnormal{Op}[(x,[\xi])\mapsto\sigma_{R_{p}}(x,[\xi])\widehat{\mathcal{M}}_{(Q+\varepsilon)p}(\xi)]f(x)\|_{L^{2}}$
$\displaystyle\leq\sup_{\|f\|_{L^{2}}=1}\|\sup_{z\in
G}|\textnormal{Op}[[\xi]\mapsto\sigma_{R_{p}}(z,[\xi])\widehat{\mathcal{M}}_{(Q+\varepsilon)p}(\xi)]f(x)|\|_{L^{2}}$
$\displaystyle\leq\sup_{\|f\|_{L^{2}}=1}\sum_{|\alpha|\leq{[n/2]+1}}\|\textnormal{Op}[[\xi]\mapsto
X_{z}^{\alpha}\sigma_{R_{p}}(z,[\xi])\widehat{\mathcal{M}}_{(Q+\varepsilon)p}(\xi)]f(x)\|_{L^{2}(G_{x}\times
G_{z})}$
$\displaystyle\leq\sup_{\|f\|_{L^{2}}=1}\sum_{|\alpha|\leq{[n/2]+1}}\sup_{z\in
G}\|\textnormal{Op}[[\xi]\mapsto
X_{z}^{\alpha}\sigma_{R_{p}}(z,[\xi])\widehat{\mathcal{M}}_{(Q+\varepsilon)p}(\xi)]\|_{\mathscr{B}(L^{2})}\|f\|_{L^{2}}$
$\displaystyle=\sum_{|\alpha|\leq{[n/2]+1}}\sup_{z\in
G}\sup_{[\xi]\in\widehat{G}}\|X_{z}^{\alpha}\sigma_{R_{p}}(z,[\xi])\widehat{\mathcal{M}}_{(Q+\varepsilon)p}(\xi)\|_{\textnormal{op}}$
$\displaystyle\lesssim_{n}\sup_{|\alpha|\leq{[n/2]+1}}\sup_{(z,[\xi])\in
G\times\widehat{G}}\|X_{z}^{\alpha}\sigma_{R_{p}}(z,[\xi])\widehat{\mathcal{M}}_{(Q+\varepsilon)p}(\xi)\widehat{\mathcal{M}}^{\rho-\varepsilon
p}(\xi)\|_{\textnormal{op}}\|\widehat{\mathcal{M}}^{-\rho+\varepsilon
p}(\xi)\|_{\textnormal{op}}$ $\displaystyle\lesssim 1,$

where we have used that

$\sup_{(z,[\xi])\in
G\times\widehat{G}}\|X_{z}^{\alpha}\sigma_{R_{p}}(z,[\xi])\widehat{\mathcal{M}}_{(Q+\varepsilon)p}(\xi)\widehat{\mathcal{M}}^{\rho-\varepsilon
p}(\xi)\|_{\textnormal{op}}=O(1),\textnormal{
}\|\widehat{\mathcal{M}}^{-\rho+\varepsilon p}(\xi)\|_{\textnormal{op}}\leq
1,$

in view of (3.13) and the fact that $-\rho+\varepsilon p<0$. The analysis
above, and the positivity of the operator ${\mathcal{M}}_{-(Q+\varepsilon)p},$
imply that

$\displaystyle|\lim_{p\rightarrow 1^{+}}(p-1)\textnormal{\bf{Tr}}(R_{p})|$
$\displaystyle=\lim_{p\rightarrow 1^{+}}(p-1)|\textnormal{\bf{Tr}}(R_{p})|$
$\displaystyle\leq\lim_{p\rightarrow
1^{+}}(p-1)C\|R_{p}{\mathcal{M}}_{(Q+\varepsilon)p}\|_{\mathscr{B}(L^{2}(G))}|\textnormal{\bf{Tr}}({\mathcal{M}}_{-(Q+\varepsilon)p})|$
$\displaystyle\lesssim\lim_{p\rightarrow
1^{+}}(p-1)|\textnormal{\bf{Tr}}({\mathcal{M}}_{-(Q+\varepsilon)p})|$
$\displaystyle=\lim_{p\rightarrow
1^{+}}(p-1)\textnormal{\bf{Tr}}({\mathcal{M}}^{-(Q+\varepsilon)p})$
$\displaystyle=\textnormal{\bf{Tr}}_{\omega}({\mathcal{M}}^{-(Q+\varepsilon)}).$

Because of Lemma 3.1, we have that
$\textnormal{\bf{Tr}}_{\omega}({\mathcal{M}}^{-(Q+\varepsilon)})=0,$ and then

$\lim_{p\rightarrow 1^{+}}(p-1)\textnormal{\bf{Tr}}(R_{p})=0.$

Consequently,

$\displaystyle\textnormal{\bf{Tr}}_{\omega}(A)=\lim_{p\rightarrow
1^{+}}(p-1)\left(\textnormal{\bf{Tr}}[\textnormal{Op}(\sigma_{-Q}^{p})]+\textnormal{\bf{Tr}}[R_{p}]\right)=\int\limits_{G}\|\sigma_{-Q}(x,\cdot)\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}dx.$

Thus, we end the proof by applying Lemma 2.23. ∎

Now, we will use the functional calculus to remove the positivity assumptions
in Lemma 3.3.

######  Theorem 3.4.

Let $A\in\Psi^{-Q,\mathcal{L}}_{\rho,0}(G\times\widehat{G})$ be an
$\mathcal{L}$-elliptic pseudo-differential operator and let $0<\rho\leq 1$.
Assume that the symbol of $A$ admits an asymptotic expansion

$\sigma_{A}(x,[\xi])\sim\sum_{k=-\infty}^{-Q}\sigma_{k}(x,[\xi]),\,$

in components with decreasing order, which means that, for any
$N\in\mathbb{N},$

$\sigma_{A}(x,[\xi])-\sum_{k=-N-Q}^{-Q}\sigma_{k}(x,[\xi])\in
S^{-(N+1)\rho-Q}_{\rho,0}(G\times\widehat{G}).$

Then $|A|\in\mathcal{L}^{(1,\infty)}(L^{2}(G)),$

$\textnormal{\bf{Tr}}_{\omega}(|A|)=\int\limits_{G}\|\sigma_{-Q}(x,[\xi])\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}dx,$

and we have

$\textnormal{Det}_{\omega,|A|}(\lambda)=\exp\left(\textnormal{\bf{Tr}}_{\omega}(|A|)\lambda\right)=\exp\left(\int\limits_{G}\|\sigma_{-Q}(x,[\xi])\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}dx\cdot\lambda\right),$
(3.14)

for $\lambda\in\mathbb{C}$ with $|\lambda|$ small enough, and

$\textnormal{Det}_{\omega,|A|}^{\prime}(0):=\frac{d}{d\lambda}\textnormal{Det}_{\omega,|A|}(\lambda)|_{\lambda=0}=\textnormal{\bf{Tr}}_{\omega}(|A|).$

###### Proof.

From the subelliptic calculus (see (i) in Theorem 2.11), we have that the
matrix-valued symbol $\sigma_{A^{*}A}$ of $A^{*}A,$ admits the asymptotic
expansion

$\sigma_{A^{*}A}(x,[\xi])\sim\sum_{k=-\infty}^{-2Q}\tilde{\sigma}_{k}(x,[\xi]),\,$

in components with decreasing order, which means that, for any
$N\in\mathbb{N},$

$\sigma_{A}(x,[\xi])-\sum_{k=-N-2Q}^{-2Q}{\tilde{\sigma}}_{k}(x,[\xi])\in
S^{-(N+1)\rho-2Q}_{\rho,0}(G\times\widehat{G}),$

with the principal term given by

${\tilde{\sigma}}_{-2Q}(x,[\xi])={\sigma}_{-Q}(x,[\xi])^{*}{\sigma}_{-Q}(x,[\xi])=|{\sigma}_{-Q}(x,[\xi])|^{2},$

and consequently positive for any $(x,[\xi]).$ The functional calculus in
[CR20], implies that the symbol of $|A|:=\sqrt{A^{*}A},$ admits an asymptotic
expansion of the type

$\sigma_{|A|}(x,[\xi])\sim\sum_{k=-\infty}^{-Q}{\sigma^{\prime}}_{k}(x,[\xi]),\,$

that is, for any $N\in\mathbb{N},$

$\sigma_{|A|}(x,[\xi])-\sum_{k=-N-Q}^{-Q}{\sigma^{\prime}}_{k}(x,[\xi])\in
S^{-(N+1)\rho-Q}_{\rho,0}(G\times\widehat{G}),$

with the principal term given by

${\sigma^{\prime}}_{-Q}(x,[\xi])=\sqrt{{\sigma}_{-Q}(x,[\xi])^{*}{\sigma}_{-Q}(x,[\xi])}=|{\sigma}_{-Q}(x,[\xi])|,$

which is then positive for any $(x,[\xi]).$ Applying Lemma 3.3 to $|A|,$ and
observing that

$\|\sigma^{\prime}_{-Q}(x,[\xi])\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}=\|\sigma_{-Q}(x,[\xi])\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})},$

we conclude that

$\textnormal{\bf{Tr}}_{\omega}(|A|)=\int\limits_{G}\|\sigma_{-Q}(x,[\xi])\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}dx.$

We end the proof by applying Lemma 2.23. ∎

In the case of a positive subelliptic operator we can compute its Dixmier
trace as follows.

######  Corollary 3.5.

Let $A\in\Psi^{-Q,\mathcal{L}}_{\rho,0}(G\times\widehat{G})$ be a positive
$\mathcal{L}$-elliptic pseudo-differential operator and let $0<\rho\leq 1$.
Assume that the symbol of $A$ admits an asymptotic expansion

$\sigma_{A}(x,[\xi])\sim\sum_{k=-\infty}^{-Q}\sigma_{k}(x,[\xi]),\,$

in components with decreasing order, which means that, for any
$N\in\mathbb{N},$

$\sigma_{A}(x,[\xi])-\sum_{k=-N-Q}^{-Q}\sigma_{k}(x,[\xi])\in
S^{-(N+1)\rho-Q}_{\rho,0}(G\times\widehat{G}).$ (3.15)

Then $A\in\mathcal{L}^{(1,\infty)}(L^{2}(G))$ and its Dixmier trace can be
computed as

$\textnormal{\bf{Tr}}_{\omega}(A)=\int\limits_{G}\|\sigma_{-Q}(x,[\xi])\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}dx.$
(3.16)

Furthermore, we have

$\textnormal{Det}_{\omega,A}(\lambda)=\exp\left(\textnormal{\bf{Tr}}_{\omega}(A)\lambda\right)=\exp\left(\int\limits_{G}\|\sigma_{-Q}(x,[\xi])\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}dx\cdot\lambda\right),$
(3.17)

for $\lambda\in\mathbb{C}$ with $|\lambda|$ small enough,

$\textnormal{Det}_{\omega,A}^{\prime}(0):=\frac{d}{d\lambda}\textnormal{Det}_{\omega,A}(\lambda)|_{\lambda=0}=\textnormal{\bf{Tr}}_{\omega}(A),$

and the determinant formula

$\textnormal{\bf
Det}_{\omega}(I+A)=\exp\left(\int\limits_{G}\|\sigma_{-Q}(x,[\xi])\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}dx\right)$

holds true.

###### Proof.

Since $A$ is positive, we can apply Theorem 3.4 to $A=|A|$ and the proof can
be concluded. ∎

######  Corollary 3.6.

Under the hypothesis of Lemma 3.3, we have

$\textnormal{\bf{Tr}}_{\omega}(A)=\int\limits_{G}\|\sigma_{-Q}(x,[\xi])\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}dx=\lim_{N\rightarrow\infty}\frac{1}{\log(N)}\sum_{j=1}^{N}s_{j}(A),$
(3.18)

where $s_{j}(A)$ denotes the sequence of singular values of $A$ with
multiplicities taken into account.

###### Proof.

It follows from the Dixmier Theorem 2.19 that $\textnormal{\bf
Tr}_{\omega}(A)=\textnormal{\bf Tr}_{\textnormal{Dix}}(A),$ provided that the
limit $\textnormal{\bf Tr}_{\textnormal{Dix}}(A)$ exists. However, the
existence of the limit $\lim_{p\rightarrow 1^{+}}(p-1)\textnormal{\bf
Tr}(A^{p}),$ is equivalent to the existence of $\textnormal{\bf
Tr}_{\textnormal{Dix}}(A),$ [Con94, Page 313] and in the case of their
existence they have the same value, that is $\textnormal{\bf Tr}_{\omega}(A)$.
The identity in (3.18) now follows from the proof of Lemma 3.3, where we have
shown that $\textnormal{\bf Tr}_{\omega}(A)=\lim_{p\rightarrow
1^{+}}(p-1)\textnormal{\bf Tr}(A^{p}).$ ∎

If in Corollary 3.5 we replace the sub-Laplacian $\mathcal{L}$ by the Laplace-
Beltrami operator on $G,$ then $Q=n=\dim(G)$ and we obtain the following
result for the standard Hörmander classes on $G,$ in view of the equivalence
of classes
$\Psi^{-n}_{\rho,0}(G\times\widehat{G})=\Psi^{-n}_{\rho,0}(G;\textnormal{loc})$
for all $0<\rho\leq 1.$

######  Corollary 3.7.

Let $A\in\Psi^{-n}_{\rho,0}(G,\textnormal{loc})$ with $0<\rho\leq 1,$ be a
positive elliptic pseudo-differential operator on $G$. Assume that the symbol
of $A$ admits an asymptotic expansion

$\sigma_{A}(x,[\xi])\sim\sum_{k=-\infty}^{-n}\sigma_{k}(x,[\xi]),\,$ (3.19)

in components with decreasing order, which means that, for any
$N\in\mathbb{N},$

$\sigma_{A}(x,[\xi])-\sum_{k=-N-n}^{-n}\sigma_{k}(x,[\xi])\in
S^{-(N+1)\rho-n}_{\rho,0}(G\times\widehat{G}).$ (3.20)

Then $A\in\mathcal{L}^{(1,\infty)}(L^{2}(G))$ and

$\textnormal{\bf
Tr}_{\omega}(A)=\int\limits_{G}\|\sigma_{-n}(g,[\xi])\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}dg.$
(3.21)

In view of the Connes equivalence Theorem 2.26 and the equivalence of classes
$\Psi^{-n}_{1,0}(G\times\widehat{G})=\Psi^{-n}_{1,0}(G;\textnormal{loc})$
proved in [RTW14] we have:

######  Corollary 3.8.

Let $A\in\Psi^{-n}_{cl}(G,\textnormal{loc})$ be a positive and elliptic
pseudo-differential operator on $G$. Assume that the symbol of $A$ admits an
asymptotic expansion

$\sigma_{A}(x,[\xi])\sim\sum_{k=-\infty}^{-n}\sigma_{k}(x,[\xi]),\,$ (3.22)

in components with decreasing order, which means that, for any
$N\in\mathbb{N},$

$\sigma_{A}(x,[\xi])-\sum_{k=-N-n}^{-n}\sigma_{k}(x,[\xi])\in
S^{-(N+1)-n}_{1,0}(G\times\widehat{G}).$ (3.23)

Then the Wodzicki residue of $A$ can be computed according to the formula

$\textnormal{res}(A)=\int\limits_{G}\|\sigma_{-n}(x,[\xi])\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}dx.$
(3.24)

Now, we present the main theorem of this section.

######  Theorem 3.9.

Let $A\in\Psi^{-Q,\mathcal{L}}_{\rho,0}(G\times\widehat{G})$ be an
$\mathcal{L}$-elliptic pseudo-differential operator and let $0<\rho\leq 1$.
Assume that the symbol of $A$ admits an asymptotic expansion

$\sigma_{A}(x,[\xi])\sim\sum_{k=-\infty}^{-Q}\sigma_{k}(x,[\xi]),\,$

in components with decreasing order, which means that, for any
$N\in\mathbb{N},$

$\sigma_{A}(x,[\xi])-\sum_{k=-N-Q}^{-Q}\sigma_{k}(x,[\xi])\in
S^{-(N+1)\rho-Q}_{\rho,0}(G\times\widehat{G}).$

Then $A\in\mathcal{L}^{(1,\infty)}(L^{2}(G)),$ and its Dixmier trace is given
by

$\displaystyle\textnormal{\bf{Tr}}_{\omega}(A)=\int\limits_{G}\left(\|\textnormal{Re}(\sigma_{-Q}(x,[\xi]))^{+}\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}-\|\textnormal{Re}(\sigma_{-Q}(x,[\xi]))^{-}\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}\right)dx$
$\displaystyle+i\int\limits_{G}\left(\|\textnormal{Im}(\sigma_{-Q}(x,[\xi]))^{+}\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}-\|\textnormal{Im}(\sigma_{-Q}(x,[\xi]))^{-}\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}\right)dx.$

###### Proof.

In view of the polar decomposition $A=|A|U$ of $A,$ where $U$ is a unitary
operator on $L^{2}(G)$ (see e.g. [GGK00]), that
$\mathcal{L}^{(1,\infty)}(L^{2}(G))$ is an ideal of operators in the
$C^{*}$-algebra $\mathscr{B}(L^{2}(G)),$ that
$|A|\in\mathcal{L}^{(1,\infty)}(L^{2}(G))$ in view of Theorem 3.4, we deduce
that $A\in\mathcal{L}^{(1,\infty)}(L^{2}(G)).$ Now we will use the linearity
properties of the Dixmier trace in order to compute the Dixmier trace of $A.$

We use the decomposition of $A$ into its real and imaginary part as in Remark
2.20,

$\textnormal{Re}(A):=\frac{A+A^{*}}{2},\,\,\textnormal{Im}(A):=\frac{A-A^{*}}{2i},$

and the decomposition of $\textnormal{Re}(A)$ and $\textnormal{Im}(A)$ into
their positive and negative parts,

$\displaystyle\textnormal{Re}(A)^{+}:=\frac{\textnormal{Re}(A)+|\textnormal{Re}(A)|}{2},\,\,\textnormal{Re}(A)^{-}:=\frac{|\textnormal{Re}(A)|-\textnormal{Re}(A)}{2},$

and

$\displaystyle\textnormal{Im}(A)^{+}:=\frac{\textnormal{Im}(A)+|\textnormal{Im}(A)|}{2},\,\,\textnormal{Im}(A)^{-}:=\frac{|\textnormal{Im}(A)|-\textnormal{Im}(A)}{2}.$

Now, the operator $A$ can be written as

$\displaystyle A$ $\displaystyle=\textnormal{Re}(A)+i\textnormal{Im}(A)$
$\displaystyle=\left(\textnormal{Re}(A)^{+}-\textnormal{Re}(A)^{-}\right)+i\left(\textnormal{Im}(A)^{+}-\textnormal{Im}(A)^{-}\right).$

So, by the linearity of the Dixmier trace $\textnormal{Tr}_{\omega}$ we have

$\displaystyle\textnormal{Tr}_{\omega}(A)$
$\displaystyle=\textnormal{Tr}_{\omega}(\textnormal{Re}(A))+i\textnormal{Tr}_{\omega}(\textnormal{Im}(A))$
$\displaystyle=\left(\textnormal{Tr}_{\omega}(\textnormal{Re}(A)^{+})-\textnormal{Tr}_{\omega}(\textnormal{Re}(A)^{-})\right)+i\left(\textnormal{Tr}_{\omega}(\textnormal{Im}(A)^{+})-\textnormal{Tr}_{\omega}(\textnormal{Im}(A)^{-})\right).$

Now, we will exploit the subelliptic functional calculus in [CR20] in order to
compute the symbols of the positive operators
$\textnormal{Re}(A)^{+},\textnormal{Re}(A)^{-},\textnormal{Im}(A)^{+},\textnormal{Im}(A)^{-}.$
Indeed, we have

$\displaystyle\sigma_{\textnormal{Re}(A)}(x,[\xi])=\sigma_{\frac{A+A^{*}}{2}}(x,[\xi])=\textnormal{Re}(\sigma_{-Q}(x,[\xi]))+\textnormal{lower
order terms},$
$\displaystyle\sigma_{\textnormal{Im}(A)}(x,[\xi])=\sigma_{\frac{A-A^{*}}{2i}}(x,[\xi])=\textnormal{Im}(\sigma_{-Q}(x,[\xi]))+\textnormal{lower
order terms},$
$\displaystyle\sigma_{\textnormal{Re}(A)^{+}}(x,[\xi])=\sigma_{\frac{\textnormal{Re}(A)+|\textnormal{Re}(A)|}{2}}(x,[\xi])=\textnormal{Re}(\sigma_{-Q}(x,[\xi]))^{+}+\textnormal{lower
order terms},$
$\displaystyle\sigma_{\textnormal{Re}(A)^{-}}(x,[\xi])=\sigma_{\frac{|\textnormal{Re}(A)|-\textnormal{Re}(A)}{2}}(x,[\xi])=\textnormal{Re}(\sigma_{-Q}(x,[\xi]))^{-}+\textnormal{lower
order terms},$

and

$\displaystyle\sigma_{\textnormal{Im}(A)^{+}}(x,[\xi])=\sigma_{\frac{\textnormal{Im}(A)+|\textnormal{Im}(A)|}{2}}(x,[\xi])=\textnormal{Im}(\sigma_{-Q}(x,[\xi]))^{+}+\textnormal{lower
order terms},$
$\displaystyle\sigma_{\textnormal{Im}(A)^{-}}(x,[\xi])=\sigma_{\frac{|\textnormal{Im}(A)|-\textnormal{Im}(A)}{2}}(x,[\xi])=\textnormal{Im}(\sigma_{-Q}(x,[\xi]))^{-}+\textnormal{lower
order terms}.$

Now, by applying Corollary 3.5, we can eliminate the lower terms when
computing the Dixmier trace, and we have the following formulae

$\textnormal{\bf{Tr}}_{\omega}(\textnormal{Re}(A)^{+})=\int\limits_{G}\|\textnormal{Re}(\sigma_{-Q}(x,[\xi]))^{+}\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}dx$
$\textnormal{\bf{Tr}}_{\omega}(\textnormal{Re}(A)^{-})=\int\limits_{G}\|\textnormal{Re}(\sigma_{-Q}(x,[\xi]))^{-}\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}dx$
$\textnormal{\bf{Tr}}_{\omega}(\textnormal{Im}(A)^{+})=\int\limits_{G}\|\textnormal{Im}(\sigma_{-Q}(x,[\xi]))^{+}\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}dx$
$\textnormal{\bf{Tr}}_{\omega}(\textnormal{Im}(A)^{-})=\int\limits_{G}\|\textnormal{Im}(\sigma_{-Q}(x,[\xi]))^{-}\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}dx.$

In view of the linearity of the Dixmier trace $\textnormal{\bf{Tr}}_{\omega}$
and the previous trace formula we end the proof. ∎

Now, let us replace the sub-Laplacian $\mathcal{L}$ by the Laplace-Beltrami
operator on $G$ in Theorem 3.9. As mentioned above, we obtain $Q=n=\dim(G)$
and we obtain the following result for the standard Hörmander classes on $G,$
in view of the equivalence of classes
$\Psi^{-n}_{\rho,0}(G\times\widehat{G})=\Psi^{-n}_{\rho,0}(G;\textnormal{loc})$
for all $0<\rho\leq 1,$ see [RTW14].

######  Corollary 3.10.

Let $A\in\Psi^{-n}_{\rho,0}(G,\textnormal{loc})$ with $0<\rho\leq 1,$ be an
elliptic pseudo-differential operator on $G$. Assume that the symbol of $A$
admits an asymptotic expansion

$\sigma_{A}(x,[\xi])\sim\sum_{k=-\infty}^{-n}\sigma_{k}(x,[\xi]),\,$ (3.25)

in components with decreasing order, which means that, for any
$N\in\mathbb{N},$

$\sigma_{A}(x,[\xi])-\sum_{k=-N-n}^{-n}\sigma_{k}(x,[\xi])\in
S^{-(N+1)\rho-n}_{\rho,0}(G\times\widehat{G}).$ (3.26)

Then $A\in\mathcal{L}^{(1,\infty)}(L^{2}(G))$ and its Dixmier trace is given
by

$\displaystyle\textnormal{\bf{Tr}}_{\omega}(A)=\int\limits_{G}\left(\|\textnormal{Re}(\sigma_{-Q}(x,[\xi]))^{+}\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}-\|\textnormal{Re}(\sigma_{-Q}(x,[\xi]))^{-}\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}\right)dx$
$\displaystyle+i\int\limits_{G}\left(\|\textnormal{Im}(\sigma_{-Q}(x,[\xi]))^{+}\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}-\|\textnormal{Im}(\sigma_{-Q}(x,[\xi]))^{-}\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}\right)dx.$

In view of the Connes equivalence Theorem 2.26 and the equivalence of classes
$\Psi^{-n}_{1,0}(G\times\widehat{G})=\Psi^{-n}_{1,0}(G;\textnormal{loc})$ we
have:

######  Corollary 3.11.

Let $A\in\Psi^{-n}_{cl}(G,\textnormal{loc})$ be an elliptic pseudo-
differential operator on $G$. Assume that the symbol of $A$ admits an
asymptotic expansion

$\sigma_{A}(x,[\xi])\sim\sum_{k=-\infty}^{-n}\sigma_{k}(x,[\xi]),\,$ (3.27)

in components with decreasing order, which means that, for any
$N\in\mathbb{N},$

$\sigma_{A}(x,[\xi])-\sum_{k=-N-n}^{-n}\sigma_{k}(x,[\xi])\in
S^{-(N+1)-n}_{1,0}(G\times\widehat{G}).$ (3.28)

Then the Wodzicki residue of $A$ can be computed according to the formula

$\displaystyle\textnormal{res}(A)=\int\limits_{G}\left(\|\textnormal{Re}(\sigma_{-Q}(x,[\xi]))^{+}\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}-\|\textnormal{Re}(\sigma_{-Q}(x,[\xi]))^{-}\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}\right)dx$
$\displaystyle+i\int\limits_{G}\left(\|\textnormal{Im}(\sigma_{-Q}(x,[\xi]))^{+}\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}-\|\textnormal{Im}(\sigma_{-Q}(x,[\xi]))^{-}\|_{\mathcal{L}^{(1,\infty)}(\widehat{G})}\right)dx.$

##  4\. Final remarks on compact manifolds

We now illustrate some connections between the determinant of operators in the
setting of the invariance notion recalled in Section 2.4 and introduced in
[DR18], and the problem of computing the Dixmier trace of elliptic operators.
We denote by $\Psi^{m}_{+e}(M)$ the class of positive elliptic classical
pseudo-differential operators of order $m\in\mathbb{R}.$

If $A\in\Psi^{-{n}}_{+e}(M)$, then $A$ is not necessarily an operator of trace
class. Indeed, the best information that we know about the Schwartz kernel
$K_{A}$ of $A,$ is its logarithmic behaviour over the diagonal (see Hörmander
[Hör85]). So, the determinant function $\textnormal{Det}(I+\lambda\cdot)$ is
not a well defined functional on the class of operators $\Psi^{-{n}}_{+e}(M)$.
However, when studying the regularised version

$\textnormal{Det}_{\omega,A}(\lambda):=\lim_{p\rightarrow
1^{+}}\textnormal{Det}(I+\lambda A^{p})^{p-1},$ (4.1)

of $`\textnormal{Det}(I+\lambda A)$’, we can observe that it can be computed
in terms of the global symbol
$\sigma_{A,E}\equiv(\sigma_{A,E}(\ell))_{\ell\in\mathbb{N}}$ of $A,$ and of
its Dixmier trace (and also, in view of the Connes equivalence in Theorem
2.26, of its non-commutative residue).

######  Remark 4.1.

Let $M$ be a compact manifold of dimension $n$ without boundary endowed with a
volume form $dx$, and let $E$ be a positive elliptic classical pseudo-
differential operator of order $\nu>0.$ Let us denote the eigenvalues of $E$
by $\\{\lambda_{j}\\}_{j\geq 0}.$ If $A:C^{\infty}(M)\rightarrow
C^{\infty}(M)$ is invariant with respect to the subspaces
$H_{j}=\textnormal{Ker}(E-\lambda_{j}I),$ in terms of the symbol
$\sigma_{A,E}(\ell)$ of $A,$ let us define the functional

$\|\sigma_{A,E}\|_{\mathcal{L}^{(1,\infty)}(\mathbb{N},E)}:=\frac{1}{n}\lim_{N\rightarrow\infty}\frac{1}{\log
N}\sum_{l:(1+\lambda_{l})^{\frac{1}{\nu}}\leq
N}\textnormal{\bf{Tr}}(|\sigma_{A,E}(l)|).$ (4.2)

It was proved in [CDC20], that $A\in\mathcal{L}^{(1,\infty)}(L^{2}(M)),$ if
and only if,

$\|\sigma_{A,E}\|_{\mathcal{L}^{(1,\infty)}(\mathbb{N},E)}<\infty.$

In particular, if $A$ is positive (with respect to the inner product on
$L^{2}(M)$),

$\textnormal{\bf{Tr}}_{\omega}(A)=\|\sigma_{A,E}\|_{\mathcal{L}^{(1,\infty)}(\mathbb{N},E)}.$

In view of Lemma 2.23 and Remark 4.1 we have the following theorem.

######  Theorem 4.2.

Let $A\in\Psi^{-n}_{+e}(M)$ be a positive and elliptic operator. Then
$A\in\mathcal{L}^{(1,\infty)}(L^{2}(M)),$ and

$\textnormal{Det}_{\omega,A}(\lambda)=\exp\left(\textnormal{\bf{Tr}}_{\omega}(A)\lambda\right)=\exp\left(\lambda\|\sigma_{A,E}\|_{\mathcal{L}^{(1,\infty)}(\mathbb{N},E)}\right),$
(4.3)

for $\lambda\in\mathbb{C}$ with $|\lambda|$ small enough and any
$E\in\Psi^{-n}_{+e}(M)$ commuting with $A$. In particular,

$\textnormal{Det}_{\omega,A}^{\prime}(0):=\frac{d}{d\lambda}\textnormal{Det}_{\omega,A}(\lambda)|_{\lambda=0}=\textnormal{\bf{Tr}}_{\omega}(A)=\|\sigma_{A,E}\|_{\mathcal{L}^{(1,\infty)}(\mathbb{N},E)},$

and

$\textnormal{\bf{Det}}_{\omega}(I+A)=\exp(\|\sigma_{A,E}\|_{\mathcal{L}^{(1,\infty)}(\mathbb{N},E)}).$
(4.4)

###### Proof.

Let us prove that $A$ satisfies the hypothesis in Lemma 2.23. Indeed, the fact
that $A\in\Psi^{-n}_{+}(M),$ implies that
$A\in\mathcal{L}^{(1,\infty)}(L^{2}(M))$ (see [Con94]). The functional
calculus for elliptic operators implies that $A^{p}\in\Psi^{-np}_{+}(M),$ and
consequently, for $1<p<\infty,$ $\textnormal{\bf{Tr}}_{\omega}(A^{p})=0.$
Indeed, the condition $A^{p}\in\Psi^{-np}_{+}(M),$ implies that $A^{p}\in
S_{1}(L^{2}(M))$ which implies that its Dixmier functional vanishes (see
[Con94]). In view of Lemma 2.23 we have proved the left hand side of (4.3).
The right-hand side of (4.3) is a consequence of Remark 4.1. ∎

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