# Irreducible representations of $\mathrm{SL}(2,\mathbb{R})$
on Hilbert spaces are admissible

Francesca Astengo  Dipartimento di Matematica, Dipartimento di Eccellenza
2023–2027, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy
<EMAIL_ADDRESS>,  Michael G. Cowling  School of Mathematics and
Statistics, University of New South Wales, Sydney NSW 2052, Australia
<EMAIL_ADDRESS>and  Bianca Di Blasio  Dipartimento di Matematica e
Applicazioni, Università di Milano Bicocca, Via Cozzi 53, 20125 Milano, Italy
<EMAIL_ADDRESS>
###### Abstract.

We show that irreducible strongly continuous representations of
$\mathrm{SL}(2,\mathbb{R})$ on Hilbert spaces are admissible, modulo the
recently proposed solution of the invariant subspace problem on Hilbert
spaces.

###### Key words and phrases:

###### 1991 Mathematics Subject Classification:

Primary: 22E46; Secondary: 22E45, 22E47

The second author was supported by the Australian Research Council, Grant
DP220100285. He also thanks the Universities of Genova and Milano Bicocca for
their hospitality.

##  1\. Introduction

Throughout this paper, $G$ stands for the noncompact semisimple Lie group
$\mathrm{SL}(2,\mathbb{R})$ and $K$ for its compact subgroup $\mathrm{SO}(2)$.
An irreducible representation $\rho$ of $G$ on a Banach space $\mathcal{X}$ is
said to be _admissible_ if $\rho|_{K}$, its restriction to $K$, decomposes as
a topological direct sum of characters of $K$, in which no character appears
more than once. We are going to prove the following result.

######  Theorem 1.1.

Assume that the invariant subspace problem for Hilbert spaces has a positive
solution. Then every irreducible representation of $G$ on a Hilbert space is
admissible.

We give the proof towards the end of this note. Our analysis focusses on the
Banach algebra $L^{1}(G,\omega)_{m}$ of functions $f$ with the invariance and
integrability properties

$f(k_{\theta}xk_{\phi})=e^{-im(\theta+\phi)}f(x)\qquad\text{and}\qquad\int_{G}\left\|x\right\|^{\alpha}\left|f(x)\right|\,dx<\infty$

(the notation will be explained in Section 2 below, together with other
background material). In Section 3 we review the theory of $m$-spherical
functions on $G$ and the associated spherical transformation; our key
technical result is a theorem that ensures that functions with the
$K$-equivariance property above and whose $m$-spherical transforms behave well
belong to $L^{1}(G,\omega)_{m}$. Section 4 uses spectral theory and the
results of Sections 2 and 3 to show that there exists $f_{m}\in
L^{1}(G,\omega)_{m}$ that generates all $L^{1}(G,\omega)_{m}$; this result is
in the spirit of some work of R. Gangolli [14]. Finally in Section 5 we use
the function $f_{m}$ of Section 4 and the putative solution of the invariant
subspace problem on Hilbert space to prove our main theorem.

In the rest of this introduction, we explain the significance of Theorem 1.1.
In the early period of the development of the representation theory of
semisimple Lie groups, between 1945 and 1975, say, representations on Hilbert
spaces and on Banach spaces were considered; see, for instance, [24, Chap 4].
It was shown that irreducible _unitary_ representations on Hilbert spaces are
admissible, which implies in particular that they may be effectively studied
and classified by passing to the Lie algebra of the group in question, but the
situation with Banach representations was unclear, though it was known that
there were some pathological examples of reducible Banach representations (see
below for an example). In 1988, W. Soergel [22] finally constructed an example
of an irreducible representation of $G$ on a Banach space that was not
admissible.

Soergel’s construction used the resolution of the invariant subspace problem.
This problem, whether every nontrivial operator on a complex Banach space of
dimension greater than $1$ admits a nontrivial invariant subspace, was first
resolved in the negative by P. Enflo [9], and later shorter solutions were
produced by C.J. Read [19, 20]. All the examples produced were operators on
Banach spaces that are rather different from Hilbert spaces, so it was not
clear whether Soergel’s construction would apply to Hilbert space
representations or not. What was clear, however, was that if the invariant
subspace problem has a positive solution on a Banach space $\mathcal{X}$, then
pathological examples could be constructed on the Banach space of a
representation induced from a representation of $\mathbb{R}$ on $\mathcal{X}$.
Hence while we have a relatively simple classification of all the irreducible
unitary representations, for general Banach space representations, a similar
classification is probably impossible.

In the last year, two solutions have been proposed to the invariant subspace
problem on Hilbert spaces, by Enflo [10] and by C. W. Neville [18], both
affirming that every bounded operator on a Hilbert space of dimension greater
than $1$ _does_ admit a nontrivial invariant subspace. We do not propose to
confirm the correctness of these papers here. Rather, the aim of this note is
to take this proposed solution as a starting point and to then show that it
implies that every irreducible representation of $G$ on a Hilbert space _is_
admissible. At this point, it is not clear whether the invariant subspace
problem has a positive or negative solution when the Banach space under
consideration is “better than reflexive but worse than Hilbertian”, for
instance if $\mathcal{X}$ is an $L^{p}$ space where $1<p<\infty$. However, our
arguments, coupled with those of Soergel [22], imply that the admissibility of
representations of $\mathrm{SL}(2,\mathbb{R})$ and the existence of a positive
solution of the invariant subspace problem are essentially equivalent.

Together with our previous paper [2], our result also shows that a
classification of irreducible uniformly bounded representations of $G$ on
Hilbert spaces is possible; every such representation is equivalent (via
conjugation by a bounded operator with bounded inverse) to one of the
irreducible uniformly bounded representations described by R. A. Kunze and E.
M. Stein [16], and the Kunze–Stein representations are minimal representatives
of the equivalence classes.

##  2\. Definitions and preliminary results

The early work on $\mathrm{SL}(2,\mathbb{R})$ is due to V. Bargmann [3]. L.
Ehrenpreis and F. Mautner [6, 7, 8] used Bargmann’s work later to study
analysis on $\mathrm{SL}(2,\mathbb{R})$ in detail. Most of what we write in
this section would have been known to these authors.

### The group $G$

Recall that $G=\mathrm{SL}(2,\mathbb{R})$ and $K=\mathrm{SO}(2)$. The Iwasawa
decomposition is that every $x$ in $G$ may be written uniquely in the form
$n_{t}a_{r}k_{\theta}$, where $r,t\in\mathbb{R}$ and $\theta\in\mathbb{R}\mod
2\pi$; here

$\begin{gathered}k_{\theta}:=\begin{pmatrix}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{pmatrix}=\exp\left(\theta\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\right)\\\ a_{r}:=\begin{pmatrix}e^{r}&0\\\
0&e^{-r}\end{pmatrix}=\exp\left(r\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\right)\\\ n_{t}:=\begin{pmatrix}1&0\\\
t&1\end{pmatrix}=\exp\left(t\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}\right)\
.\end{gathered}$

According to the Cartan decomposition, every element of $G$ may be written in
the form $k_{\theta}a_{r}k_{\phi}$, where $r\geq 0$; if $x\in K$ then $r=0$
and there are many choices for $k_{\theta}$ and $k_{\phi}$; otherwise, the
decomposition is unique up to changes in $\theta$ and $\phi$ by adding (or
subtracting) $\pi$ to both or $2\pi$ to either. In these coordinates,

(2.1) $dx=2\sinh(2r)\,dr\,d\varphi\,d\psi.$

Clearly

$2\sinh(2r)\sim\begin{cases}4r&\text{as $r\to 0$}\\\ e^{2r}&\text{as
$r\to\infty$}.\end{cases}$

The Lie algebra $\mathfrak{g}$ of $G$ is spanned by the elements $E$, $H$ and
$Q$, given by

(2.2) $E=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},\qquad
H=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},\qquad Q=\begin{pmatrix}0&1\\\
1&0\end{pmatrix};$

we define the Casimir operator $\Omega$ by

$4\Omega:=H^{2}+Q^{2}-E^{2}.$

### Representations of $G$

Throughout, we use the word _representation_ to mean a representation $\rho$
of $G$ on a topological vector space $\mathcal{V}$ by continuous operators
whose inverse is also continuous, which is strongly continuous, in the sense
that $x_{n}\to x$ in $G$ implies that $\rho(x_{n})\xi\to\rho(x)\xi$ in
$\mathcal{V}$ for all $\xi\in\mathcal{V}$. If $\mathcal{V}$ is a Hilbert space
or a Banach space, we call $\rho$ a Hilbert or a Banach representation. For
$x\in G$, we write $\|x\|$ for the norm of $x$ as a linear operator on the
$2$-dimensional real Hilbert space $\mathbb{R}^{2}$. It is evident that
$\|x\|=\|x^{-1}\|$ and $\|x_{1}x_{2}\|\leq\|x_{1}\|\|x_{2}\|$ for all
$x,x_{1},x_{2}$ in $G$, and that if $x$ has Cartan decomposition
$ka_{r}k^{\prime}$, then $\|x\|=\|a_{r}\|=e^{r}$. Norms in other spaces are
indicated by a subscript, for example,
$\left\|\rho(x)\right\|_{\mathcal{L}(\mathcal{X})}$ indicates the norm of
$\rho(x)$ in the Banach spaces of bounded linear operators on a Banach space
$\mathcal{X}$.

######  Lemma 2.1.

Suppose that $\rho$ is a representation of $G$ on a Banach space
$\mathcal{X}$. Then there exist constants $\alpha$ in $[0,\infty)$ and $C$ in
$[1,\infty)$ such that

(2.3) $\|\rho(x)\|_{\mathcal{L}(\mathcal{X})}\leq C\|x\|^{\alpha}\qquad\forall
x\in G.$

###### Proof.

First, $x\mapsto\|\rho(x)\xi\|_{\mathcal{X}}$ is continuous on $G$, and so is
bounded on compact sets. The Banach–Steinhaus theorem implies that
$\|\rho(x)\|_{\mathcal{L}(\mathcal{X})}$ is bounded on compact sets in $G$.

If $j\in\mathbb{N}^{+}$ and $\|x\|\leq e^{j}$, then we may write $x=x_{1}\dots
x_{j}$, where $\|x_{i}\|\leq e$; indeed, if $x=ka^{r}k^{\prime}$, then we take
$x_{1}$ to be $ka_{r/j}$, $x_{2}$, …, $x_{j-1}$ to be $a_{r/j}$, and $x_{j}$
to be $a_{r/j}k^{\prime}$. Hence

$\|\rho(x)\|_{\mathcal{L}(\mathcal{X})}\leq\|\rho(x_{1})\|_{\mathcal{L}(\mathcal{X})}\dots\|\rho(x_{j})\|_{\mathcal{L}(\mathcal{X})}\leq\|a_{r/j}\|_{\mathcal{L}(\mathcal{X})}^{j}\leq\max\\{\|\rho(x)\|_{\mathcal{L}(\mathcal{X})}:\|x\|\leq
e\\}^{j};$

the result follows with $C$ equal to
$\max\\{\|\rho(x)\|_{\mathcal{L}(\mathcal{X})}:\|x\|\leq e\\}$ and $\alpha$
equal to $\log C$. ∎

A representation satisfying (2.3) is said to be of growth rate $\alpha$.

We now consider the restriction to $K$ of a representation $\rho$ of $G$ on a
Banach space $\mathcal{X}$. For $m\in\mathbb{Z}$, we define the closed
subspace $\mathcal{X}_{m}$ of $\mathcal{X}$ by

$\mathcal{X}_{m}:=\\{\xi\in\mathcal{X}:\rho(k_{\theta})\xi=e^{im\theta}\xi\\},$

and $\mathcal{P}_{m}:\mathcal{X}\to\mathcal{X}_{m}$ by

$\mathcal{P}_{m}:=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-im\theta}\rho(k_{\theta})\,d\theta.$

Then $\mathcal{P}_{m}$ is a bounded projection; indeed,

$\left\|\mathcal{P}_{m}\right\|_{\mathcal{L}(\mathcal{X})}\leq\frac{1}{2\pi}\int_{-\pi}^{\pi}\left\|\rho(k_{\theta})\right\|_{\mathcal{L}(\mathcal{X})}\,d\theta\leq\max\\{\left\|\rho(k_{\theta})\right\|_{\mathcal{L}(\mathcal{X})}:k_{\theta}\in
K\\}.$

.

### The algebra $L^{1}(G,\omega)_{m}$

Let $L^{1}(G)_{m}$ and $L^{2}(G)_{m}$ be the subsets of $L^{1}(G)$ and
$L^{2}(G)$ of functions $f$ such that

(2.4) $f(k_{\theta}xk_{\phi})=e^{-im(\theta+\phi)}f(x)\qquad\forall
k_{\theta},k_{\phi}\in K\quad\forall x\in G.$

The spaces $L^{1}(G)_{m}$ and $L^{2}(G)_{m}$ are obviously closed subspaces of
$L^{1}(G)$ and $L^{2}(G)$.

######  Lemma 2.2.

The space $L^{1}(G)_{m}$, equipped with convolution and the $L^{1}(G)$-norm,
is a commutative Banach algebra.

###### Proof.

See [17, Lemma 3, p. 21] or [23, Proposition 1, p. 290]. ∎

For $\alpha\in[0,\infty)$, let $\omega_{\alpha}$, or just $\omega$, be the
function $x\mapsto\|x\|^{\alpha}$. Then

$\omega(x)\in[1,\infty),\qquad\omega(x)=\omega(x^{-1})\qquad\text{and}\qquad\omega(x_{1}x_{2})\leq\omega(x_{1})\omega(x_{2})$

for all $x,x_{1},x_{2}\in G$. A continuous function satisfying these
conditions is called a weight. We write $L^{1}(G,\omega)$ for the subset of
$L^{1}(G)$ of functions $f$ such that

$\left\|f\right\|_{L^{1}(G,\omega)}:=\int_{G}\left|f(x)\right|\omega(x)\,dx<\infty.$

Then $L^{1}(G,\omega)$ is a subspace of $L^{1}(G)$ that is closed under
convolution, and equipped with its own norm, is a Banach algebra (often called
a Beurling algebra). Indeed,

$\displaystyle\left\|f*g\right\|_{L^{1}(G,\omega)}$
$\displaystyle\leq\int_{G}\left|f\right|*\left|g\right|(x)\omega(x)\,dx$
$\displaystyle\leq\int_{G}\left|f(y)\right|\left|g(y^{-1}x)\right|\omega(y)\omega(y^{-1}x)\,dy\,dx$
$\displaystyle=\left\|f\right\|_{L^{1}(G,\omega)}\left\|g\right\|_{L^{1}(G,\omega)}.$

Evidently $C^{\infty}_{c}(G)$ is a subspace of $L^{1}(G,\omega)$, no matter
what weight is considered. We add the subscript $m$ to both $L^{1}(G,\omega)$
and $C^{\infty}_{c}(G)$ to indicate the subspaces of functions that satisfy
(2.4). These are subalgebras of the convolution algebra $L^{1}(G)_{m}$.

The continuous multiplicative linear functionals on the commutative algebras
$C^{\infty}_{c}(G)_{m}$ were determined by Ehrenpreis and Mautner [7], and are
called $m$-spherical functions. Every continuous multiplicative linear
functional on $L^{1}(G,\omega)_{m}$ restricts to a continuous multiplicative
linear functional on the dense subspace $C^{\infty}_{c}(G)_{m}$, and hence is
the continuous extension to $L^{1}(G,\omega)$ of one of the known
$m$-spherical functions. The multiplicative linear functionals on the Banach
algebras $L^{1}(G)_{m}$, with no weight, were first studied by Ehrenpreis and
Mautner [8, Lemma 7.2], and further analysed in [21, 23]. We will recall some
properties of $m$-spherical functions and extend some of the results of these
authors to $L^{1}(G,\omega)_{m}$ in Section 3.

The algebras $L^{1}(G,\omega)_{m}$ do not have an identity, but the larger
space $M(G,\omega)_{m}$ of suitably bounded complex measures with the (measure
theoretic version of the) invariance property (2.4) contains the measure
$\mu_{m}$:

$\mu_{m}(f)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(k_{\theta})e^{-im\theta}\,d\theta,$

which has the properties that $\mathcal{P}_{m}=\rho(\mu_{m})$ and

$\mu_{m}*f=f*\mu_{m}=f\qquad\forall f\in L^{1}(G,\omega)_{m}.$

If $(f_{n})_{n\in\mathbb{N}}$ is an approximate identity in the algebra
$C^{\infty}_{c}(G)$, then $(\mu_{m}*f_{n}*\mu_{m})_{n\in\mathbb{N}}$ is an
approximate identity in the algebra $L^{1}(G,\omega)_{m}$ that consists of
smooth compactly supported functions. By considering approximate identities
whose support shrinks down to the set $K$, we may show that the norm of the
convolution operator $g\mapsto f*g$ on $L^{1}(G,\omega)_{m}$ is equal to
$\left\|f\right\|_{L^{1}(G,\omega)_{m}}$.

### Irreducible representations

By Lemma 2.1, a Banach representation $\rho$ of $G$ has a growth rate
$\alpha$. We take $\omega(x)$ to be $\|x\|^{\alpha}$, and then $\rho$ induces
a representation, also denoted by $\rho$, of $L^{1}(G,\omega)_{m}$ on
$\mathcal{X}_{m}$ by integration. Since $L^{1}(G,\omega)_{m}$ contains an
approximate identity of smooth functions, the subspace
$\mathcal{X}_{m}^{\infty}$ of all smooth vectors in $\mathcal{X}_{m}$ is dense
in $\mathcal{X}_{m}$. The space $\mathcal{X}_{m}^{\infty}$ is also the
subspace of $\mathcal{X}^{\infty}$, the space of smooth vectors in
$\mathcal{X}$, of vectors satisfying (2.4).

######  Lemma 2.3.

Let $\rho$ be an irreducible representation of $G$ on the Banach space
$\mathcal{X}$. Then the restriction of $\rho$ to $L^{1}(G,\omega)_{m}$ is an
irreducible representation of $L^{1}(G,\omega)_{m}$ on $\mathcal{X}_{m}$.

###### Proof.

To prove our claim, it suffices to take $\xi$ in
$\mathcal{X}_{m}\setminus\\{0\\}$ and $\eta$ in $\mathcal{X}_{m}$ and show
that there is a sequence $(h_{n})_{n\in\mathbb{N}}$ in $L^{1}(G,\omega)_{m}$
such that $\rho(h_{n})\xi\to\eta$ as $n\to\infty$.

Since $\rho$ is an irreducible representation of $G$, its integrated version
is an irreducible representation of $L^{1}(G,\omega)$ and there exists a
sequence $(g_{n})_{n\in\mathbb{N}}$ in $L^{1}(G,\omega)$ such that
$\lim_{n}\rho(g_{n})\xi=\eta$ in $\mathcal{X}$. Then
$h_{n}:=\mu_{m}*g_{n}*\mu_{m}\in L^{1}(G,\omega)_{m}$, and

$\eta=\mathcal{P}_{m}\eta=\lim_{n}\mathcal{P}_{m}\rho(g_{n})\xi=\lim_{n}\mathcal{P}_{m}\rho(g_{n})\mathcal{P}_{m}\xi=\lim_{n}\rho(\mu_{m}*g_{n}*\mu_{m})\xi,$

so we are done. ∎

### Admissible representations

The representation $\rho$ of $G$ gives rise to a representation of
$\mathfrak{g}$ on $\sum_{m\in\mathbb{Z}}\mathcal{X}_{m}^{\infty}$ (the
algebraic sum): for $X\in\mathfrak{g}$ and
$\xi\in\sum_{m\in\mathbb{Z}}\mathcal{X}_{m}^{\infty}$,

$\rho(X)\xi=\frac{d}{dt}\rho(\exp(tX))\xi|_{t=0};$

we extend $\rho$ to a representation of $\mathfrak{g}$ by linearity.

We say that $\rho$ is an admissible representation when each $\mathcal{X}_{m}$
is finite dimensional. This means that $\mathcal{X}_{m}$ coincides with
$\mathcal{X}_{m}^{\infty}$, and the algebraic direct sum
$\sum_{m\in\mathbb{Z}}\mathcal{X}_{m}$ is then a space on which both
$\mathfrak{g}$ and $K$ act, with an obvious compatibility condition between
the two actions. When an additional finite generation condition is satisfied,
which is automatic for an irreducible representation, such a space is known as
a Harish-Chandra module. Conversely, every Harish-Chandra module arises from a
representation (though this representation need not be irreducible). Indeed,
the Casselman–Wallach theorem shows that there is always a continuous Fréchet
space representation of $G$ associated to any Harish-Chandra module, and P.
Delorme and S. Souaifi [5] later showed that we may take the Fréchet space to
be a Hilbert space; see [4] for more information.

### Pathological representations

There are natural representations of $G$ acting on $C(T)$ and $L^{\infty}(T)$,
the spaces of continuous and bounded measurable functions on the circle $T$.
It is easy to construct pathological representations of $G$, which are weakly
but not strongly continuous, by considering the quotient representation on
$L^{\infty}(T)/C(T)$.

The next lemma shows that irreducible Banach representations are not too
pathological.

######  Lemma 2.4.

Suppose that $\rho$ is an irreducible representation of $G$ on a Banach space
$\mathcal{X}$. Then $\mathcal{X}$ is separable.

###### Proof.

Take $\xi\in\mathcal{X}$. The closure of the $G$-invariant subspace
$\\{\rho(f)\xi:f\in L^{1}(G,\omega)\\}$ of $\mathcal{X}$ is $G$-invariant and
separable, and must coincide with $\mathcal{X}$ if $\rho$ is irreducible. ∎

##  3\. The generalised spherical transformation

Consider the subalgebra $C^{\infty}_{c}(G)_{m}$ of the commutative Banach
algebra $L^{1}(G)_{m}$. The multiplicative linear functionals on
$C^{\infty}_{c}(G)_{m}$ are given by integration against generalised spherical
functions. We denote by $\phi_{m,s}$, where $s\in\mathbb{C}$, the
(generalised) $m$-spherical function such that

$\phi_{m,s}(x):=\int_{K}f_{m,s}(k^{-1}xk)\,dk,$

where $f_{m,s}(k_{\theta}a_{r}n_{t})=e^{-im\theta}\,e^{r(1-2s)}$.

It is easily checked that
$|\phi_{m,s}|\leq\phi_{0,\left|\operatorname{Re}s\right|}$, and that

$\Omega\phi_{m,s}=\left(s^{2}-\frac{1}{4}\right)\phi_{m,s}.$

The (generalised) $m$-spherical transform of $f$ in $C^{\infty}_{c}(G)_{m}$ is
defined to be the function $\hat{f}$ given by

(3.1) $\hat{f}(s):=\int_{G}f(x)\,\phi_{m,s}(x^{-1})\,dx\qquad\forall
s\in\mathbb{C}.$

Spherical functions corresponding to the same eigenvalues are equal, so
$\phi_{m,s}=\phi_{m,-s}$, and therefore $\hat{f}(s)=\hat{f}(-s)$. It has been
observed by many authors [3, 15, 21] that the $m$-spherical functions may be
expressed in terms of hypergeometric functions or Jacobi functions. Indeed,

(3.2) $\displaystyle\phi_{m,s}(a_{r})$ $\displaystyle=(\cosh
r)^{-m}\,{{}_{2}F_{1}}\left(\tfrac{1-m}{2}-s,\tfrac{1-m}{2}+s;1;-\sinh^{2}r\right)$
$\displaystyle=(\cosh
r)^{-1-2s}\,{{}_{2}F_{1}}\left(\tfrac{1+m}{2}+s,\tfrac{1-m}{2}+s;1;\tanh^{2}r\right)$
$\displaystyle=(\cosh
r)^{-1+2s}\,{{}_{2}F_{1}}\left(\tfrac{1-m}{2}-s,\tfrac{1+m}{2}-s;1;\tanh^{2}r\right).$

These identities imply that
$\phi_{m,s}(a_{r})=\phi_{-m,s}(a_{r})=\phi_{m,-s}(a_{r})$.

When $\delta\geq 0$, define the closed vertical strip $\mathbb{S}_{\delta}$
and the finite set $\mathbb{D}_{m}$ in the complex plane by

$\mathbb{S}_{\delta}:=\left\\{s\in\mathbb{C}\,:\,|\operatorname{Re}(s)|\leq\delta/2\right\\}$

and

$\mathbb{D}_{m}:=\left\\{s\in\mathbb{R}\setminus\\{0\\}\,:\,2|s|\in|m|-1-2\mathbb{N}\right\\}\
\qquad\text{where $\mathbb{N}=\\{0,1,\ldots\\}$}.$

When $s\in\mathbb{D}_{m}$, depending on the sign of $s$, the hypergeometric
function in the second or third line of (3.2) is a polynomial of degree
$(|m|-1)/2-|s|$. Moreover, as $r\to\infty$

$|\phi_{m,s}(a_{r})|\lesssim\begin{cases}(1+r)\,e^{-r}&\text{ if }s=0\\\
e^{(2|\operatorname{Re}(s)|-1)r}&\text{ if
}s\in\mathbb{C}\setminus\mathbb{D}_{m},s\neq 0\\\ e^{(-2|s|-1)r}&\text{ if
}s\in\mathbb{D}_{m}\ .\end{cases}$

Take $\omega(x)$ to be $|x|^{\alpha}$. Then
$x\mapsto|\phi_{m,s}(x)|\,\omega^{-1}(x)$ is bounded when $s$ is in
$\mathbb{S}_{\alpha+1}\cup\mathbb{D}_{m}$ and therefore we can extend the map
$f\mapsto\hat{f}(s)$ continuously to $L^{1}(G,\omega)_{m}$. If $f\in
L^{1}(G,\omega)_{m}$ has compact support, then $\hat{f}$ is an entire function
and for all $\delta>0$

(3.3) $\sup_{s\in\mathbb{S}_{\delta}}\bigl{|}\hat{f}(s)\bigr{|}\leq
C_{\delta}.$

.

Finally, we note that if $s\in\mathbb{D}_{m}$, then $\phi_{m,s}$ is in
$L^{2}(G)_{m}$, and if $s\in\mathbb{D}_{m}\setminus\mathbb{S}_{\alpha+1}$,
then $\phi_{m,s}$ is in $L^{1}(G,\omega)_{m}$. Indeed,

(3.4)
$\|\phi_{m,s}\|_{L^{1}(G,\omega)}\lesssim\int_{0}^{\infty}e^{-2|s|-1}\,e^{\alpha
r}\,e^{2r}\,dr\leq\frac{1}{2|s|-\alpha-1}.$

The asymptotic behaviour of the spherical functions for large $r$ was studied
in [15, p. 53] and [21, Lemma 4.7]. Define the (generalised) Harish-Chandra
function $\mathbf{c}_{m}$ by the rule

$\displaystyle\mathbf{c}_{m}(s):=\frac{2^{1-2s}\,\Gamma\left(2s\right)}{\Gamma\left(\frac{1+2s+m}{2}\right)\Gamma\left(\frac{1+2s-m}{2}\right)}\
.$

Then the function $\mathbf{c}_{m}^{-1}$ is meromorphic with simple poles at
the points $s$ with

$2s\in-|m|-1-2\mathbb{N}\qquad\text{or}\qquad 0<2s\in|m|-1-2\mathbb{N}\ .$

Moreover

(3.5) $|\mathbf{c}_{m}(s)^{-1}|\sim|s|^{1/2}\qquad\text{as $s\to\infty$ in
$\mathbb{S}_{\delta}$}.$

When $2s$ is not an integer, using [11, 2.10 (3)], we may write for large $r$

(3.6)
$\phi_{m,s}(a_{r})=\mathbf{c}_{m}(s)\,\Phi_{m,s}(a_{r})+\mathbf{c}_{m}(-s)\,\Phi_{m,-s}(a_{r}),$

where

$\displaystyle\Phi_{m,s}(a_{r})$ $\displaystyle=(2\cosh
r)^{2s-1}\,{{}_{2}F_{1}}\left(\tfrac{1-m}{2}-s,\tfrac{1+m}{2}-s;1-2s;\cosh^{-2}r\right)$
$\displaystyle=e^{(2s-1)r}(1+\nu_{m}(s,r)),$

and where the function $s\mapsto\nu_{m}(s,r)$ is holomorphic in
$\operatorname{Re}(s)<1$ and, for all fixed $\sigma>0$, satisfies

(3.7) $\sup_{\operatorname{Re}(s)\leq 0}|\nu_{m}(s,r)|\leq C_{m}\qquad\forall
r\geq\sigma.$

The positive-definite $m$-spherical functions were determined in [23]. The
spherical functions with $s$ in $\mathbb{D}_{m}$ correspond to the
representations in the discrete series.

The following inversion formulas for $f$ in $C^{\infty}_{c}(G)_{m}$ are proved
in [23, 15]:

(3.8)
$f(x)=\frac{1}{8\pi^{2}}\int_{\mathbb{R}}\hat{f}(i\lambda)\,\phi_{m,i\lambda}(x)\,|\mathbf{c}_{m}(i\lambda)|^{-2}\,d\lambda+\frac{1}{8\pi}\sum_{s\in\mathbb{D}_{m}}|s|\,\hat{f}(s)\,\phi_{m,s}(x);$

and if $\delta\geq 0$, $\delta\notin\mathbb{D}_{m}$, $x\not=e$, then

(3.9)
$f(x)=\frac{1}{4\pi^{2}}\int_{\mathbb{R}}\hat{f}(\delta+i\lambda)\,\mathbf{c}_{m}(\delta+i\lambda)^{-1}\,\Phi_{-\delta-i\lambda}(x)\,d\lambda+\frac{1}{8\pi}\sum_{s\in\mathbb{D}_{m}\setminus\mathbb{S}_{2\delta}}|s|\,\hat{f}(s)\,\phi_{m,s}(x).$

Formula (3.8), which is well known, can be easily extended to $f\in
L^{1}(G)_{m}$ such that

$\int_{\mathbb{R}}|\hat{f}(i\lambda)|\,|\mathbf{c}_{m}(i\lambda)|^{-2}\,d\lambda<\infty.$

Koornwinder proved formula (3.9) for $\delta>m/2$ and stated that (3.8)
follows. We show now how (3.8) implies (3.9) because we need this argument for
the proof of Theorem 3.1.

Notice that $\mathbf{c}_{m}(-i\lambda)=\overline{\mathbf{c}_{m}(i\lambda)}$
for every $\lambda\in\mathbb{R}$. Hence by (3.6) and (3.8), since $\hat{f}$ is
even, when $r>0$ we obtain

$\displaystyle f(a_{r})$
$\displaystyle=\frac{1}{4\pi^{2}}\int_{\mathbb{R}}\hat{f}(i\lambda)\,\mathbf{c}_{m}(-i\lambda)\,\Phi_{m,-i\lambda}(a_{r})\,|\mathbf{c}_{m}(i\lambda)|^{-2}\,d\lambda+\frac{1}{8\pi}\sum_{s\in\mathbb{D}_{m}}|s|\,\hat{f}(s)\,\phi_{m,s}(a_{r})$
$\displaystyle=\frac{1}{4\pi^{2}}\int_{\mathbb{R}}\hat{f}(i\lambda)\,\mathbf{c}_{m}(i\lambda)^{-1}\,\Phi_{m,-i\lambda}(a_{r})\,d\lambda+\frac{1}{8\pi}\sum_{s\in\mathbb{D}_{m}}|s|\,\hat{f}(s)\,\phi_{m,s}(a_{r}).$

Let $\delta\in(0,+\infty)\setminus\mathbb{D}_{m}$. We shift the contour of
integration, taking care of the simple poles of the function
$\mathbf{c}_{m}^{-1}$, whose residue at $s_{j}=\frac{|m|-1}{2}-j>0$ is
$(-1)^{j}\binom{|m|-j}{j}\frac{1}{2^{2-|m|+2j}}$. Combining this fact with
[12, 10.8 (16)], we obtain

$\mathrm{Res}(\mathbf{c}_{m}^{-1}(s)\Phi_{m,-s},s=s_{j})=\frac{s_{j}}{2}\,\phi_{m,s_{j}}\
.$

Applying the Cauchy theorem gives (3.9).

Here is our multiplier theorem. The reader who is interested in sharp results
will notice that the conditions here are stronger than necessary.

######  Theorem 3.1.

Let $\delta>\alpha+1$ and
$\psi\colon\mathbb{S}_{\delta}\cup\mathbb{D}_{m}\longrightarrow\mathbb{C}$ be
even and holomorphic in $\operatorname{Int}\mathbb{S}_{\delta}$. If

(3.10)
$M_{\delta}(\psi)=\sup_{s\in\mathbb{S}_{\delta}}\,(1+|s|)^{3}|\psi(s)|<\infty,$

then there exists a unique function $f\in L^{1}(G,\omega)_{m}$ such that
$\hat{f}=\psi$, and

$\|f\|_{L^{1}(G,\omega)}\leq
C_{m}\,\Bigl{(}M_{\delta}(\psi)+\sum_{s\in\mathbb{D}_{m}\setminus\mathbb{S}_{\delta}}\frac{|s\,\psi(s)|}{2|s|-\alpha-1}\,\,\Bigr{)}.$

###### Proof.

According to the inversion formula (3.8), we take

$f(x)=\frac{1}{8\pi^{2}}\int_{\mathbb{R}}\psi(i\lambda)\,\phi_{m,i\lambda}(x)\,|\mathbf{c}_{m}(i\lambda)|^{-2}\,d\lambda+\frac{1}{8\pi}\sum_{s\in\mathbb{D}_{m}}|s|\,\psi(s)\,\phi_{m,s}(x)\
,$

and let $\chi$ be the characteristic function of $\\{x\in G\,:\,\|x\|\leq
1\\}$.

Since spherical functions are locally bounded and $|s\,\psi(s)|\leq
M_{\delta}(\psi)$ if $s\in\mathbb{D}_{m}\cap\mathbb{S}_{\delta}$, from (3.5)
we obtain

$\displaystyle|f(x)|$
$\displaystyle\leq\frac{1}{8\pi^{2}}\,M_{\delta}(\psi)\,\int_{\mathbb{R}}\,|\lambda|\,(1+|\lambda|)^{-3}\,\,d\lambda+\frac{1}{8\pi}\sum_{s\in\mathbb{D}_{m}}|s\,\psi(s)|$
$\displaystyle\leq
C\Bigl{(}M_{\delta}(\psi)+\sum_{s\in\mathbb{D}_{m}\setminus\mathbb{S}_{\delta}}|s\,\psi(s)|\,\Bigr{)},$

so $f$ is bounded when $\|x\|\leq 1$, and $\|f\chi\|_{L^{1}(G,\omega)}$
satisfies the required norm estimate.

We now estimate $|f(x)|$ when $\|x\|$ is large. Since $f$ is of type $m$, it
suffices to consider $|f(a_{r})|$ for large $r$.

Let $t\in(\frac{\alpha+1}{2},\frac{\delta}{2})\setminus\mathbb{D}_{m}$. Our
hypotheses on $\psi$ allow us to apply Cauchy’s theorem as before, and we
obtain

$f(a_{r})=\frac{1}{4\pi^{2}}\,\int_{\mathbb{R}}\psi(t+i\lambda)\,\mathbf{c}_{m}(t+i\lambda)^{-1}\,\Phi_{m,-t-i\lambda}(a_{r})\,d\lambda+\frac{1}{8\pi}\sum_{s\in\mathbb{D}_{m}\setminus\mathbb{S}_{2t}}|s|\,\psi(s)\,\phi_{m,s}(a_{r}).$

Recall from (3.4) that the spherical functions $\phi_{m,s}$ with
$s\in\mathbb{D}_{m}\setminus\mathbb{S}_{\alpha+1}$ lie in
$L^{1}(G,\omega)_{m}$, so the finite sum is easily handled. Moreover, by (3.5)
and (3.7),

$\displaystyle\left|\int_{\mathbb{R}}\psi(t+i\lambda)\,\mathbf{c}_{m}(t+i\lambda)^{-1}\,\Phi_{m,-t-i\lambda}(a_{r})\,d\lambda\right|$
$\displaystyle\qquad=e^{-(1+2t)r}\,\left|\int_{\mathbb{R}}\psi(t+i\lambda)\,\mathbf{c}_{m}(t+i\lambda)^{-1}\,(1+\nu_{m}(-t-i\lambda,r))\,e^{-2i\lambda
r}\,d\lambda\right|$ $\displaystyle\qquad\leq
C_{m}\,e^{-(1+2t)r}\,\int_{\mathbb{R}}|\psi(t+i\lambda)|\,(1+|\lambda|)^{1/2}\,d\lambda$
$\displaystyle\qquad\leq C_{m}\,e^{-(1+2t)r}\,M_{\delta}(\psi).$

The theorem follows by integration in Cartan coordinates:

$\displaystyle\|f(1-\chi)\|_{L^{1}(G,\omega)_{m}}$ $\displaystyle\leq
C_{m}\,M_{\delta}(\psi)\,\int_{1}^{\infty}e^{-(1+2t)r}\,\sinh(2r)\,e^{\alpha
r}\,dr$ $\displaystyle\leq
C_{m}\,M_{\delta}(\psi)\,\int_{1}^{\infty}e^{-(2t-\alpha-1)r}\,dr,$

as required. ∎

##  4\. Spectral theory and the spherical transformation

Let $\Omega_{m}$ be the restriction of the differential operator $\Omega$ to
$L^{2}(G)_{m}$. We fix a weight $\omega_{\alpha}$, where
$\alpha\in[0,\infty)$, and define $\alpha_{m}$ to be an arbitrary real number
strictly greater than
$\tfrac{1}{4}\max\left\\{\alpha^{2}+2\alpha,m^{2}-2|m|\right\\}$.

######  Theorem 4.1.

There exists a “heat semigroup” $(h_{t})_{t\in\mathbb{R}^{+}}$ of functions in
$L^{1}(G,\omega)_{m}$ that satisfies the conditions

$h_{s}*h_{t}=h_{s+t}\qquad\text{and}\qquad\|h_{t}\|_{L^{1}(G,\omega)}\lesssim
e^{\alpha_{m}t}$

for all $s,t\in\mathbb{R}^{+}$, and, setting $u(\cdot,t)=f*h_{t}$ for $f\in
L^{2}(G)_{m}$,

$\frac{\partial}{\partial
t}u(\cdot,t)+\Omega_{m}u(\cdot,t)=0\qquad\text{and}\qquad u(\cdot,t)\to
f\quad\text{as $t\to 0$}.$

###### Proof.

We define $h_{t}\in L^{2}(G)_{m}$ by its spherical transform:

$\hat{h}_{t}(s):=\exp(t(s^{2}-\tfrac{1}{4}))\qquad\forall s\in\mathbb{C}.$

If $s=u+iv$, then $|\hat{h}_{t}(u+iv)|=\exp(t(u^{2}-v^{2}-1/4))$, so
$\hat{h}_{t}$ is bounded and decays exponentially as we approach infinity in
every strip $\mathbb{S}_{\delta}$. Theorem 3.1 ensures that $h_{t}\in
L^{1}(G,\omega)$. The semigroup property and the differential equation follow
from the Fourier analysis.

It now suffices to show the norm estimate, for once this is established, the
convergence of $u(\cdot,t)$ as $t\to 0$ also follows. The norm estimate
follows from Theorem 3.1:

$\displaystyle\|h_{t}\|_{L^{1}(G,\omega)_{m}}$ $\displaystyle\leq
C\Bigl{(}\sup_{s\in\mathbb{S}_{\delta}}\,(1+|s|)^{3}|e^{t(s^{2}-1/4)}|+\sum_{s\in\mathbb{D}_{m}\setminus\mathbb{S}_{\delta}}\frac{|s|e^{t(s^{2}-1/4)}}{2|s|-\alpha-1}\Bigr{)}$
$\displaystyle\leq
C^{\prime}\Bigl{(}e^{t(\delta^{2}-1)/4}+\max_{s\in\mathbb{D}_{m}\setminus\mathbb{S}_{\delta}}|e^{t(s^{2}-1/4)}|\Bigr{)},$

as claimed. ∎

By improving the multiplier theorem, we could improve the $e^{\alpha_{m}t}$
term in Theorem 4.1 when $|m|$ is small to a polynomial term, but we do not
bother to do so here.

We define $\beta_{m}$ to be $(\alpha_{m}+1/4)^{1/2}$ and $\gamma_{m}$ to be
$(\alpha_{m}+5/4)^{1/2}$.

We are going to use Fourier analysis, and it is convenient to work in the
Fourier variable. For $z\in\mathbb{C}\setminus\mathbb{S}_{2\beta_{m}}$, we
define the function $r_{z}$ in $L^{2}(G)_{m}$ by

(4.1) $\hat{r}_{z}(s):=(z^{2}-s^{2})^{-1}\qquad\forall
s\in\operatorname{Int}\mathbb{S}_{2\beta_{m}}.$

######  Lemma 4.2.

The mapping $z\mapsto r_{z}$ of (4.1) is continuous from
$\mathbb{C}\setminus\mathbb{S}_{\gamma_{m}}$ to $L^{1}(G,\omega)_{m}$,

(4.2)
$\left\|r_{\zeta}\right\|_{L^{1}(G,\omega)}\lesssim(\zeta^{2}-\beta_{m}^{2})^{-1}\qquad\forall\zeta\in\mathbb{R}\setminus\mathbb{S}_{2\beta_{m}},$

and

(4.3) $\left\|r_{z}\right\|_{L^{1}(G,\omega)}\lesssim(1+|z|)^{4}\qquad\forall
z\in\mathbb{C}\setminus\mathbb{S}_{\gamma_{m}}.$

Further, for all $N\in\mathbb{N}$,

$\lim_{\zeta\in\mathbb{R},\
|\zeta|\to\infty}\left\|(\zeta^{2}r_{\zeta})^{*N}*f-f\right\|_{L^{1}(G,\omega)}\to
0\qquad\forall f\in L^{1}(G,\omega)_{m},$

where $f^{*N}$ denotes the $N$-fold convolution power of $f$.

###### Proof.

We shall first prove that, if $z$ and $\zeta$ are
in$\mathbb{C}\setminus\mathbb{S}_{\gamma_{m}}$, then $r_{z}-r_{\zeta}\in
L^{1}(G,\omega)_{m}$ and

(4.4)
$\left\|r_{z}-r_{\zeta}\right\|_{L^{1}(G,\omega)}\lesssim|\zeta^{2}-z^{2}|(1+|z|)^{2}(1+|\zeta|)^{2}.$

Then we shall prove (4.2). Once we have done so, we may take $\zeta$ in (4.4)
to be $\gamma_{m}$, and (4.3) follows immediately.

To prove (4.4), we apply the multiplier theorem. By definition,

$\hat{r}_{z}(s)-\hat{r}_{\zeta}(s)=\frac{1}{z^{2}-s^{2}}-\frac{1}{\zeta^{2}-s^{2}}=\frac{\zeta^{2}-z^{2}}{(z^{2}-s^{2})(\zeta^{2}-s^{2})}\,.$

There is a constant $C$ such that, for all$s\in\mathbb{S}_{\beta_{m}}$ and
$z\in\mathbb{C}\setminus\mathbb{S}_{\gamma_{m}}$,

$\left|\frac{1}{z-s}\right|\leq C\frac{1+|z|}{1+|s|}\,,$

whence

$\left|\frac{1}{(z^{2}-s^{2})(\zeta^{2}-s^{2})}\right|\leq
C^{4}\frac{(1+|z|)^{2}(1+|\zeta|)^{2}}{(1+|s|)^{4}}\,.$

Now (4.4) follows from the multiplier theorem.

To prove (4.2), we consider the operator $\Omega_{m}$, which is self-adjoint
on $L^{2}(G)_{m}$, with spectrum contained in $(-\infty,\alpha_{m}+1/4]$, and
may be treated using spectral theory. When $\lambda>\alpha_{m}+1/4$, the
resolvent operator $R_{\lambda}$ may be defined using the formula

$R_{\lambda}:=\int_{0}^{\infty}e^{-\lambda
t}e^{t\Omega_{m}}\,dt=(\lambda-\Omega_{m})^{-1}.$

Then the convolution kernel $\ell_{\lambda}$ of $R_{\lambda}$ is given by

$\ell_{\lambda}=\int_{0}^{\infty}e^{-\lambda t}h_{t}\,dt.$

By Theorem 4.1, the heat kernels $h_{t}$ lie in $L^{1}(G,\omega)_{m}$ and
$\left\|h_{t}\right\|_{L^{1}(G,\omega)}\lesssim e^{\alpha_{m}t}$ for all
$t\in\mathbb{R}^{+}$. It follows that $\ell_{\lambda}\in L^{1}(G,\omega)_{m}$
and

$\left\|\ell_{\lambda}\right\|_{L^{1}(G,\omega)}\lesssim\int_{0}^{\infty}e^{-\lambda
t}e^{\alpha_{m}t}\,dt=\frac{1}{\lambda-\alpha_{m}}\,$

for all $\lambda\in(\alpha_{m},\infty)$. Taking spherical transforms, we see
that

$\hat{\ell}_{\lambda}(s)=\int_{0}^{\infty}e^{-\lambda
t}\hat{h}_{t}(s)\,dt=\int_{0}^{\infty}e^{-\lambda
t}\exp(t(s^{2}-1/4))\,dt=(\lambda+\tfrac{1}{4}-s^{2})^{-1}.$

Comparing this formula with (4.1), we see that $r_{z}=\ell_{\lambda}$ when
$z^{2}=\lambda+\tfrac{1}{4}$. Hence if $z\in\mathbb{R}$ and$|z|>\beta_{m}$,
then $r_{z}\in L^{1}(G,\omega)_{m}$ and

$\left\|r_{z}\right\|_{L^{1}(G,\omega)}\lesssim\frac{1}{z^{2}-\beta_{m}^{2}}\,.$

The last part of the lemma is easy. For
$\zeta\in\mathbb{R}\setminus[-\gamma_{m},\gamma_{m}]$, the norms
$\left\|(\zeta^{2}r_{\zeta})^{*N}\right\|_{L^{1}(G,\omega)}$ are uniformly
bounded. Further,

$\left((\zeta^{2}r_{\zeta})^{*N}*f-f\right)\hat{\phantom{m}}(s)=\left(\frac{1}{1-s^{2}/\zeta^{2}}\right)^{N}\hat{f}(s)-\hat{f}(s).$

The multiplier theorem shows that

$\lim_{\zeta\to+\infty}\left\|(\zeta^{2}r_{\zeta})^{*N}*f-f\right\|_{L^{1}(G,\omega)}\to
0\qquad\forall f\in C^{\infty}_{c}(G)_{m}.$

By density, this extends to all $f\in L^{1}(G,\omega)_{m}$. ∎

######  Theorem 4.3.

There exists $f_{m}\in L^{1}(G,\omega)_{m}$ such that the closed subalgebra of
$L^{1}(G,\omega)_{m}$ generated by $f_{m}$ is all $L^{1}(G,\omega)_{m}$.

###### Proof.

Let $A$ be the closed subalgebra of $L^{1}(G,\omega)_{m}$ generated by the
resolvent function $r_{\gamma_{m}}$; we are going to show that
$A=L^{1}(G,\omega)_{m}$.

We recall a standard argument that the resolvent depends analytically on the
parameter $\lambda$ and the resolvent set is open. We recall that
$\mathcal{L}(\mathcal{X})$ denotes the space of bounded linear operators on
the Banach space $\mathcal{X}$.

If $R_{\lambda_{0}}\in\mathcal{L}(L^{1}(G,\omega)_{m})$, and
$|\lambda-\lambda_{0}|<1/\left\|R_{\lambda_{0}}\right\|_{\mathcal{L}(L^{1}(G,\omega)_{m})}$,
then

(4.5) $\displaystyle R_{\lambda}$
$\displaystyle=\frac{1}{\lambda-\Omega_{m}}=\sum_{n=0}^{\infty}\frac{(\lambda_{0}-\lambda)^{n}}{(\lambda_{0}-\Omega_{m})^{n+1}}=\sum_{n=0}^{\infty}(\lambda_{0}-\lambda)^{n}R^{n+1}_{\lambda_{0}},$

and this last series converges absolutely in
$\mathcal{L}(L^{1}(G,\omega)_{m})$, so
$R_{\lambda}\in\mathcal{L}(L^{1}(G,\omega)_{m})$. It also follows that when
the resolvent operator $R_{\lambda_{0}}$ is given by convolution with an
element of $L^{1}(G,\omega)_{m}$, so is $R_{\lambda}$, and when
$R_{\lambda_{0}}$ is given by convolution with an element of $A$, so is
$R_{\lambda}$. Then the set
$S:=\\{z\in\mathbb{C}\setminus\mathbb{S}_{\gamma_{m}}:r_{z}\in A\\}$ is open
in $\mathbb{C}\setminus\mathbb{S}_{\gamma_{m}}$. The set $S$ is also closed in
$\mathbb{C}\setminus\mathbb{S}_{\gamma_{m}}$, since $r_{z_{j}}\to r_{z}$ in
$L^{1}(G,\omega)_{m}$ if $z_{j}\to
z\in\mathbb{C}\setminus\mathbb{S}_{\gamma_{m}}$ by Lemma 4.2, and hence
$S=\mathbb{C}\setminus\mathbb{S}_{\gamma_{m}}$.

Take $f\in L^{1}(G,\omega)_{m}$; we must show that $f\in A$. From the Cartan
decomposition, $f$ may be approximated arbitrarily closely in
$L^{1}(G,\omega)_{m}$ by elements $f_{n}$ with compact support, and so it
suffices to show that $f_{n}\in A$. By (3.3) $\hat{f}_{n}$ is uniformly
bounded on $\mathbb{S}_{\gamma_{m}}$. By Lemma 4.2, $f_{n}$ can be
approximated arbitrarily well in $L^{1}(G,\omega)_{m}$ by elements of the form
$(\zeta^{2}r_{\zeta})^{*4}*f_{n}$ by taking $\zeta$ large in $\mathbb{R}$, so
it suffices to treat $(\zeta^{2}r_{\zeta})^{*4}*f_{n}$; it is evident that
$((\zeta^{2}r_{\zeta})^{*4}*f_{n})\hat{\phantom{m}}(s)$ vanishes like
$(1+|s|)^{-8}$ at infinity in the strip $\mathbb{S}_{\beta_{m}}$. Let
$g=(\zeta^{2}r_{\zeta})^{*4}*f_{n}$. If we can show that $g\in A$, then $f\in
A$ by a limiting argument.

The Cauchy integral formula and the evenness of $\hat{g}$ imply that

$\displaystyle\hat{g}(s)$ $\displaystyle=\frac{1}{2\pi
i}\int_{\Gamma}\hat{g}(z)\left(\frac{1}{z-s}+\frac{1}{z+s}\right)\,dz$
$\displaystyle=\frac{1}{\pi
i}\int_{\Gamma}z\hat{g}(z)\frac{1}{z^{2}-s^{2}}\,dz$
$\displaystyle=\frac{1}{\pi
i}\int_{\Gamma}z\hat{g}(z)\,\hat{r}_{z}(s)\,dz\qquad\forall
s\in\operatorname{Int}\mathbb{S}_{2\beta_{m}};$

here $\Gamma$ is the contour given by $z=\gamma_{m}+iy$ where $y$ varies from
$-\infty$ to $\infty$ in $\mathbb{R}$. Since $r_{z}\in A$ and
$\left\|r_{z}\right\|_{L^{1}(G,\omega)}\lesssim(1+|\operatorname{Im}z|)^{4}$
while $|\hat{g}(z)|\lesssim(1+|\operatorname{Im}z|)^{-8}$, this integral
converges as an $A$-valued Bochner integral, and hence $g\in A$, as required.
∎

##  5\. Irreducible Hilbert representations

Suppose that we have an irreducible Hilbert representation $\rho$ of $G$ on
$\mathcal{H}$. We shall show that $\dim(\mathcal{H}_{m})\leq 1$ for all
$m\in\mathbb{Z}$.

######  Theorem 5.1.

Assume that all bounded operators on a complex Hilbert space of dimension
greater than $1$ admit a nontrivial closed invariant subspace. If $\rho$ is an
irreducible representation of $G$ on a Hilbert space $\mathcal{H}$, then
$\dim(\mathcal{H}_{m})\leq 1$ for all $m$.

###### Proof.

Let $\rho$ be a representation of $G$ on a Hilbert space $\mathcal{H}$. As
argued in Section 2, there exists $\alpha\in[0,\infty)$ such that
$\|\rho(x)\|_{\mathcal{L}(\mathcal{H})}\leq C\|x\|^{\alpha}:=C\omega(x)$ for
all $x\in G$.

With a view to reaching a contradiction, suppose that
$\dim(\mathcal{H}_{m})>1$ for some $m\in\mathbb{Z}$. By Lemma 2.3, the
restriction to $L^{1}(G,\omega)_{m}$ of the integrated representation of
$L^{1}(G,\omega)$ acts irreducibly on $\mathcal{H}_{m}$. Take $f_{m}\in
L^{1}(G,\omega)_{m}$ that generates $L^{1}(G,\omega)_{m}$, whose existence in
proved in the previous section. Let $\mathcal{H}_{m}^{\circ}$ be a nontrivial
closed invariant subspace for $\rho(f_{m})$ acting on $\mathcal{H}_{m}$. Then
$\mathcal{H}_{m}^{\circ}$ is a nontrivial closed invariant subspace for
$\rho(L^{1}(G,\omega)_{m})$ on $\mathcal{H}_{m}$, which is our desired
contradiction. ∎

######  Corollary 5.2.

Each irreducible uniformly bounded representation $\rho$ of $G$ on a Hilbert
space is equivalent to one of the uniformly bounded representations of
Ehrenpreis and Mautner [6, 7, 8] and Kunze and Stein [16].

###### Proof.

Our theorem implies that $\rho$ is admissible. The classification of
admissible representations (see, for instance, [13, Chapter 2]) then implies
the affirmation. ∎

##  6\. Final remarks

We have seen that every irreducible Hilbert space representation of $G$ is
admissible, modulo the solution to the invariant subspace problem. Conversely,
by the embedding theorem of Delorme and Souafi [5], every irreducible Harish-
Chandra module of every seimsimple Lie group can be globalised to a Hilbert
representation. Our arguments may extend to other semisimple Lie groups.
However, the extension seems to require a version of the invariant subspace
problem that deals with families of operators, and this does not seem to have
been considered.

##  7\. Acknowledgements

The authors thank Bernhard Krötz for helpful comments.

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