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\begin{document}
\title[M\"obius transformations and monogenic functional calculus]
{M\"obius transformations and \\ monogenic functional calculus}
\author{Vladimir V. Kisil}
\address{Institute of Mathematics, %\\
                        Economics and Mechanics, %\\
                        Odessa State University, %\\
                        ul. Petra Velikogo, 2, %\\
                        Odessa-57, 270057, Ukraine}
\email{vk@imem.odessa.ua} %root@qchem.odessa.ua}

\copyrightinfo{1996}{American Mathematical Society}
\date{October 6, 1995, and, in revised form, March 9, 1996}
%\revdate{March 9, 1996}
\commby{Alexandre Kirillov}

\thanks{This work was partially supported by the INTAS grant 93-0322.
It was finished while the author enjoyed the hospitality of
Universiteit Gent, Vakgroep Wiskundige Analyse, Belgium.}
\keywords{Functional calculus, joint spectrum,  group representation,
intertwining operator, Clifford analysis, quantization} %.}
\subjclass{Primary 46H30, 47A13; Secondary 
30G35, 47A10, 47A60, 47B15, 81Q10}
\begin{abstract}
A new way of doing functional calculi is presented. A functional
calculus $\Phi: f(x)\rightarrow f(T)$ is not an algebra homomorphism
of a functional algebra into an operator algebra, but an intertwining
operator between two representations of a group acting on the two
algebras (as linear spaces).

This scheme is shown on the newly developed monogenic
functional calculus for an arbitrary set of non-commuting
self-adjoint operators. The corresponding spectrum and
spectral mapping theorem are included.
\end{abstract}
\maketitle
%\tableofcontents
\section{Introduction}

Functional calculus and the corresponding spectral theory belong to the
heart of functional
analysis~\cite{Helemskii93,SimonReed80,Vasilescu82}.
\person{J.~L.~Taylor}~\cite{JTaylor72} formulated a problem of functional
calculus in the following terms:

\textsl{For a given element $a$ of a Banach algebra $\algebra{A}$ the
correspondence $x\mapsto a$ gives rise to a representation of the
algebra of polynomials in the variable $x$ in $\algebra{A}$. It is
necessary to extend the representation to a larger functional
algebra\comment{ in the variable $x$}.}

The scheme works perfectly for several commuting variables $x_i$
and commuting elements $a_i\in\algebra{A}$, and gives rise to the
analytic Taylor calculus~\cite{JTaylor70,JTaylor72}.
But for the non-commuting case
the situation is not so simple: searching for an appropriate substitution for
a functional algebra in commuting variables was done in classes of Lie
algebras~\cite{JTaylor72}, freely generated algebras~\cite{JTaylor73},
cumbersome $\times$-algebras~\cite{KisRam95a}, etc. For the Weyl
calculus~\cite{Anderson69} no algebra homomorphism is known.

The main point of our approach (firstly presented in the paper)
is the following reformulation of the
problem, where the shift from an algebra \emph{homomorphism} to group
\emph{representations} is made:
\begin{defn}\label{de:calc}
Let a group algebra $\FSpace{L}{1}(G)$ have linear representations
${\pi}_{F}(g)$ in a space of functions $F$ and ${\pi}_B(g)$ in a Banach
algebra $B$ (depending on a set of elements $T\subset B$). One says
that a linear mapping $\Phi: F\rightarrow B$ is a {\em functional
calculus\/} if
$\Phi$ is an intertwining operator between ${\pi}_{F}$ and ${\pi}_B$,
namely
\begin{displaymath}
\Phi [{\pi}_{F}(g) f]={\pi}_B(g) \Phi[f],
\end{displaymath}
for all $g\in G$ and $f\in F$. Additional conditions will be stated later.
\end{defn}
One may ask whether there is a connection between these two
formulations at all. It is possible to check that the classic
Dunford-Riesz calculus (see~\cite[Chap.~IX]{RieszNagy55}
and~\cite[Chap.~2]{Helemskii93}) is generated
by the group of fractional-linear transformations of the complex line;
the Weyl~\cite{Anderson69} and continuous functional
calculi~\cite[Chap.~VII]{SimonReed80} are generated by the affine
group of  the real line; the analytic functional
calculus~\cite{JTaylor70,Vasilescu82} is based on
biholomorphic automorphisms of a
domain $U\subset\Space{C}{n}$;
and the monogenic (Clifford-Riesz) calculus~\cite{KisRam95a}
is generated by the group of
M\"obius transformations in $\Space{R}{n}$. Moreover, both the Weyl and
the analytic calculi can be obtained from the monogenic one by
restrictions of the symmetry group (see Theorem~\ref{th:space} and
Corollary~\ref{co:holom}). Such connections are based
on the role of group representations in function theories
(particularly on links with integral
representations~\cite{Kisil95d,Kisil95a}).

We will postpone the general results~\cite{Kisil95e} about functional
calculi. The subject of this paper is the construction of the
monogenic calculus from the M\"obius transformations (for other calculi see
Remark~\ref{re:other}). The monogenic and the
Riesz-Clifford~\cite{KisRam95a} calculi are cousins but not twins:
besides having conceptually different starting points,
they are technically different too. For example, for a
function $\Space{R}{n}\rightarrow \Cliff{n}$ we construct a calculus for
$n$-tuple of operators, not for $(n-1)$-tuples as it was done
in~\cite{KisRam95a}.

Due to the brief nature of present paper, ``proofs'' mean ``hints'' or
``sketches of proof''.

\section{Clifford analysis and the conformal group}
\comment{
In this Section we give notations and preliminary results. }

\subsection{The Clifford algebra and the M\"obius group}
Let $\Space{R}{n}$ be a Euclidean $n$-dimensional vector space with
a fixed frame $a_1$, $a_2$, \ldots, $a_n$ and let $\Cliff{n}$ be
the \emph{real Clifford algebra}~\cite[\S~12.1]{MTaylor86} generated by
$1$, $e_j$, $1\leq j\leq n$ and the relations
\begin{displaymath}
e_i e_j + e_j e_i =-2\delta_{ij}, \qquad 1e_i=e_i 1=e_i.
\end{displaymath}
We put $e_0=1$. The embedding $\algebra{i}: \Space{R}{n}\rightarrow
\Cliff{n}$ is defined by the formula:
\begin{equation}\label{eq:embedding}
\algebra{i}: x=\sum_{j=1}^n x_j a_j \mapsto \vecbf{x}=\sum_{j=1}^n x_j e_j.
\end{equation}
We identify $\Space{R}{n}$ with its image under $\algebra{i}$ and
call its elements \emph{vectors}. There are two linear
anti{-}automorphisms $*$ and $-$ of $\Cliff{n}$ defined on its basis
$A_\nu=e_{j_1}e_{j_2}\cdots e_{j_r}$, $1\leq j_1 <\cdots2$ is
not so rich as the conformal group in $\Space{R}{2}$.
Nevertheless, the conformal covariance has many applications in
Clifford analysis~\cite{Ryan95b}.
Notably,
groups of conformal mappings of open unit balls $\Space{B}{n} \subset
\Space{R}{n}$ \emph{on}to itself are similar for all $n$ and
as sets can be
parametrized by the product of $\Space{B}{n}$ itself and the group
of isometries of its boundary $\Space{S}{n-1}$.
\begin{lem}
Let $a\in\Space{B}{n}$, $b\in\Gamma_n$; then the M\"obius transformations of
the form
\begin{displaymath}
\phi_{(a,b)}=\matr{b}{0}{0}{{b}^{*-1}}\matr{1}{-
a}{{a}^*}{-1}=\matr{b}{-ba}{{b}^{*-1}a^*}
{-{b}^{*-1}}
\end{displaymath}
constitute the group $B_{n}$ of conformal mappings of the open unit ball
$\Space{B}{n}$ onto itself. $B_{n}$ acts on
$\Space{B}{n}$ transitively.
Transformations of the form $\phi_{(0,b)}$  constitute a
subgroup isomorphic to $\object{O}{(n)}$. The homogeneous space
$B_{n}/\object{O}{(n)}$ is isomorphic as a set to
$\Space{B}{n}$. Moreover:
\begin{enumerate}
\item $\phi_{(a,1)}^2=1$ identically on $\Space{B}{n}$
($\phi_{(a,1)}^{-1}=\phi_{(a,1)}$).
\item $\phi_{(a,1)}(0)=a$, $\phi_{(a,1)}(a)=0$.
\comment{
\item $\phi_{(a,1)}\phi_{(b,1)}=\phi_{(c,d)}$ where
$c=\phi_{(b,1)}(a)$ and $=\phi_{(a,1)}$ }
\end{enumerate}
\end{lem}

Obviously, conformal mappings preserve the space of harmonic
functions, i.e., null solutions to the \emph{Laplace} operator
$\Delta=\sum_{j=1}^n \frac{\partial^2 }{\partial x_j^2}$. They also
preserve the space of \emph{monogenic} functions, i.e.,
null solutions $f: \Space{R}{n}\rightarrow
\Cliff{n}$ of the \emph{Dirac}~\cite{BraDelSom82,DelSomSou92} operator $D=\sum_{j=1}^n e_i
\frac{\partial }{\partial x_j}$ (note that $\Delta=D\bar{D}=-D^2$). The
group $B_{n}$ is sufficient for construction of the Poisson
integral representation of harmonic functions and the Cauchy and
Bergman formulas in Clifford analysis by the formula~\cite{Kisil95d}
\begin{equation}\label{eq:reproduce}
K(x,y)=c\int_G [\pi_g f](x) \overline{[\pi_g f](y)}\,dg,
\end{equation}
where $\pi_g$ is an irreducible unitary
square integrable representation~\cite[\S~9.3]{Kirillov74} of a group $G$,
$f(x)$ is an arbitrary non-zero function, and $c$ is a constant.

\section{Monogenic calculus}
\subsection{Representations of the M\"obius group in \Cstar-algebras}
Fix an $n$-tuple of \emph{bounded self-adjoint} elements
$T=(T_1,\ldots,T_n)$ of a \Cstar-algebra $\algebra{A}$.  Let
${\algbf{A}}=\algebra{A}\otimes \Cliff{n}$. We can associate with $T$
an element $\vecbf{T}\in {\algbf{A}}$ by the
formula~\cite{McInPryde87}
\begin{displaymath}
\vecbf{T}=\sum_{j=1}^n T_j\otimes e_j
\end{displaymath}
in analogy with the embedding $\algebra{i}$ of~\eqref{eq:embedding}.
\comment{
We again call elements of $\algbf{A}$ in the above form by
\emph{vectors}.
We combine two natural anti{-}automorphisms $*$ and $-$ on
$\algebra{A}$ and $\Cliff{n}$ and define a linear anti-automorphism
on  ${\algbf{A}}$ by its action on elementary
tensors:
\begin{displaymath}
\overline{A\otimes a}=A^* \otimes \bar{a}, \qquad A\in \algebra{A}, \
a\in\Cliff{n}.
\end{displaymath}
Particularly  $\vecbf{T}\widetilde{\vecbf{T}}=
\widetilde{\vecbf{T}}\vecbf{T}= \sum_{j=1}^n \norm{T_j}^2$ is real we
would denote it by $\norm{\vecbf{T}}^2$. }
One notes the naturality of the following
\begin{defn}
The \emph{Clifford resolvent set} $R(\vecbf{T})$ of an $n$-tuple
$T$ is the maximal open subset of $\Space{R}{n}$ such that for
$\lambda\in R(\vecbf{T})$ the element $\vecbf{T}-\lambda I$ is invertible in
${\algbf{A}}$.
The \emph{Clifford spectrum} $\sigma(\vecbf{T})$ is the completion of the
Clifford resolvent set $\Space{R}{n}\setminus R(\vecbf{T})$.
\end{defn}
For example, the joint spectrums of one $\{j_1\}$, two $\{j_1,j_2\}$,
three $\{j_1,j_2,j_3\}$ Pauli matrices~\cite[\S~12.4]{MTaylor86} are
the unit sphere $\Space{S}{0}$ ($=\{-1,1\}$), the origin $(0,0)$ (= a sphere
of zero radius), the unit sphere $\Space{S}{2}$ in $\Space{R}{1}$,
$\Space{R}{2}$, and $\Space{R}{3}$, respectively.
The canonical connection between the Grassmann and Clifford algebras
gives (see~\cite[\S~III.6]{Vasilescu82})
\begin{lem}
Let an $n$-tuple $T$ consist of mutually commuting operators in
$\algebra{A}$. Then the Taylor joint spectrum\footnote{There are less
used non-commutative versions of the Taylor joint
spectrum~\cite{JTaylor72}.}~\cite{JTaylor70} and the Clifford spectrum
coincide.
\end{lem}
Let $V$ be a subgroup of the Vahlen group $V(n)$ such that for any
$\matr{a}{b}{c}{d}\in V$ we have $c^{-1}d \in R(\vecbf{T})$ (i.e.,
$c\vecbf{T}+d$ is invertible in $\algbf{A}$). Then there is a
(non-linear) representation $\pi_T$ of $V$:
\begin{equation}\label{eq:op-rep}
\pi_{T}\matr{a}{b}{c}{d}: \vecbf{T} \mapsto
(a\vecbf{T}+b)(c\vecbf{T}+d)^{-1}.
\end{equation}
The following easy lemma has a useful corollary.
\begin{lem}\label{le:spectr}
A vector $\lambda$ belongs to the resolvent set $R(\vecbf{T})$ if and only
if $\phi_{(\lambda,1)}(\vecbf{T})$ is well defined.
\end{lem}
\begin{cor}\label{co:spectr}
A vector $\lambda$ belongs to the spectrum $\sigma(\vecbf{T})$ if and only
if the vector $\phi_{(\lambda,1)}(\beta)$ belongs to the spectrum
$\sigma(\phi_{(\beta,1)}(\vecbf{T}))$.
\end{cor}
\comment{
\begin{defn}
An intertwining operator $\Theta$ between two representations
$\pi_{\Space{R}{n}}$~\eqref{eq:sp-rep} and $\pi_{\algebra{A}}$
is called a \emph{coordinate mapping}.
\end{defn}
}
Having a representation $\pi$ of the group $G$ in a \emph{linear} space
$X$ one can define the \emph{linear} representation $\pi_L$ of the
convolution algebra $\FSpace{L}{1}(G)$ in the space of mappings
$X\rightarrow X$ by the rule
\begin{equation}\label{eq:extension}
\pi_L(f)= \int_G f(g) \pi(g)\, dg,
\end{equation}
where $dg$ is Haar measure on $G$.

We denote such an extension of
$\pi_{\Space{R}{n}}$ of~\eqref{eq:sp-rep} by $\pi_F$ and the extension
of $\pi_{T}$ of~\eqref{eq:op-rep} by $\pi_{\algebra{A}}$.
Let $\FSpace{F}{}(\Space{R}{n})=\pi_F(\FSpace{L}{1}(G))$,
$\widehat{f}(\vecbf{x})=\pi_F(f)$,
$\widehat{f}(\vecbf{T})=\pi_\algebra{A}(f)$.
Following Definition~\ref{de:calc} we give
\begin{defn}
A \emph{monogenic functional calculus}
$\Phi_V:\FSpace{F}{}(\Space{R}{n})\rightarrow \algbf{A}$ for
a subgroup $V$ and $n$-tuple $T$ is a linear operator with the
following properties:
\begin{enumerate}
\item\label{it:fc-rep} It is an intertwining operator between the two
representations
$\pi_F$ and $\pi_{\algebra{A}}$:
\begin{displaymath}
\Phi_V \pi_{F} =\pi_{\algebra{A}} \Phi_V.
\end{displaymath}
\item\label{it:fc-iden} $\Phi \widehat{\unit}=I$, i.e., the image of the
function identically equal to $1$ is the unit operator.
\item\label{it:fc-t} $\Phi \widehat{\delta}_e=\vecbf{T}$, where $\delta_e$ is
the delta function centered at the group unit.
\end{enumerate}
\end{defn}
The following theorem is group{-}independent:
\begin{thm}\label{th:compose}
Let $k(g)$ and $l(g)$ be functions on $V$ such that
both $\widehat{l}(\vecbf{T})$ and
$\widehat{k}(\widehat{l}(\vecbf{T}))$ are well defined.
Then $\widehat{k}(\widehat{l}(\vecbf{T}))=
[\widehat{k*l}](\vecbf{T})$ ($*$ is convolution on the
group). In particular, $\pi_\algebra{A}(g)[\widehat{l}(\vecbf{T})]=
[\widehat{\pi_l(g)l}](\vecbf{T})$.
\end{thm}
\begin{thm}[Uniqueness]
If the representation $\pi_F$ is irreducible, then
the functional calculus (if any) is unique.
Particularly, for a given simply connected domain $\Omega$ and
an $n$-tuple of operators $T$, there exists no more than one
monogenic calculus.
\end{thm}
\begin{thm}\label{th:space}
For any $n$-tuple $T$ of bounded self-adjoint operators there
exists a monogenic calculus on $\Space{R}{n}$. This calculus coincides on
monogenic functions
with the Weyl functional calculus from~\cite{Anderson69}.
Particularly, the polynomial functions of operators are the symmetric
(Weyl) polynomials.
\end{thm}
\begin{proof}
M\"obius transformations of $\Space{R}{n}$ (without the point $\infty$)
are the affine transformations of $\Space{R}{n}$, which are
represented via
upper-triangular Vahlen matrices. But the Weyl calculus is defined
uniquely by its affine covariance~\cite[Theorem~2.4(a)]{Anderson69}
according to the following lemma, which is given without proof.
\end{proof}
\begin{lem}
The one-dimensional Weyl calculus is uniquely defined by its affine
covariance.
\end{lem}

The following result is a direct counterpart of the classic one.
\begin{thm}
For the existence of the monogenic functional calculus in the unit ball
$\Space{B}{n}$ it is necessary that $\sigma(\vecbf{T})\subset
\Space{B}{n}$.
\end{thm}
In the next subsection we obtain via an integral representation that this
condition is also sufficient.

\subsection{Integral representation}

The best way to obtain an integral formula for the monogenic calculus
is to use~\eqref{eq:reproduce}. To save space we will now
use  another straightforward approach, which however could be
obtained from~\eqref{eq:reproduce} by a suitable representation of
$\Space{Z}{}$.
We already know from Theorem~\ref{th:space} how to construct monogenic
functions for polynomials. Thus, by the linearity (via
decomposition~\cite[Chap.~II, (1.16)]{DelSomSou92}) there is only one
way to define the Cauchy kernel of the operators $T_j$.
\begin{defn} Let us define
(we use notation
from~\cite{DelSomSou92})\begin{equation}\label{eq:cauchy-op}
E(y,T)=\sum_{j=0}^\infty\left(\sum_{\modulus{\alpha }=j}V_\alpha
(T)W_\alpha (y)\right) ,
\end{equation}
where
\begin{align} %\begin{eqnarray}
W_\alpha (x)&=(-1)^{\modulus{\alpha}}\partial ^\alpha E(x)
=\frac{(-1)^{\modulus{\alpha}}\Gamma(\frac{n+1}{2})}{2\pi^{(n+1)/2}}\,
\partial ^\alpha \frac{\overline{x}}{\modulus{x}^{n+1}},\nonumber\\
V_\alpha(T) &= \frac{1}{\alpha !}\sum_\sigma (T_1 e_{\sigma(1)}-T_{\sigma(1)})
(T_1e_{\sigma(2)}-T_{\sigma(2)})\cdots (T_1e_{\sigma(n)}- T_{\sigma(n)}),
\label{eq:sym-pol}
\end{align} %\end{eqnarray}
where the summation is taken over all possible permutations.
\end{defn}
\begin{lem}\label{le:bound}
Let $\modulus{T}=\lim_{j\rightarrow\infty}\sup_\sigma
\norm{T_{\sigma(1)}
\cdots T_{\sigma(j)} }^{1/j}$, $1\leq\sigma(i)\leq n$ be the
Rota-Strang joint
spectral radius~\cite{Rota60a}.
Then for a fixed $\modulus{y}>\modulus{T}$,
equation~\eqref{eq:cauchy-op}
defines a bounded operator in $\algbf{A}$.
\end{lem}
As usual, by Liouville's theorem~\cite[Theorem~5.5]{McInPryde87}
for the function $E(y,T)$ the Clifford
spectrum is not empty; by definition it is closed and by the previous
lemma it is bounded. Thus
\begin{lem} The Clifford spectrum $\sigma(\vecbf{T})$ is compact.
\end{lem}
\begin{lem}\label{le:v-lem}
Let $r>\modulus{T}$ and let $\Omega$ be the ball $\Space{B}{}(0,r)\in
\Space{R}{n}$. Then \comment{for any symmetric polynomial $P(\vecbf{x})$} we
have
\begin{equation}\label{eq:int-polyn}
P(T)=\int_{\partial \Space{B}{}} E(y,T)\, dn_y\, P(y),
\end{equation}
where $P(T)$ is the symmetric polynomial~\eqref{eq:sym-pol} of the
$n$-tuple $T$.
\end{lem}
\begin{lem}
For any domain $\Omega$ that does not contain $\sigma(\vecbf{T})$ and any
$f\in\FSpace{H}{}(\Omega)$, we have
\begin{equation}
\int_{\partial \Omega} E(y,T)\, dn_y\, f(y)=0.
\end{equation}
\end{lem}

Due to this lemma we can replace the domain  $\Space{B}{}(0,r) $ in
Lemma~\ref{le:v-lem} with an arbitrary domain $\Omega$
containing the spectrum $\sigma(\vecbf{T}).$
An application of Lemma~\ref{le:v-lem} gives the
main
\begin{thm}
Let $T=(T_1,\ldots,T_n)$ be an $n$-tuple of bounded self-adjoint
operators. Let the domain $\Omega$ with piecewise smooth boundary have
a connected complement and suppose the spectrum $\sigma(\vecbf{T})$ lies
inside the domain $\Omega$. Then the mapping
\begin{equation}\label{eq:int-calc}
\Phi: f(x)\mapsto f(T)=\int_{\partial \Omega} E(y,T)\, dn_y\,
f(y)
\end{equation}
defines the monogenic calculus for $T$.
\end{thm}
Because holomorphic functions are also monogenic,
we can apply our construction to them. For a commuting
$n$-tuple $T$ the calculus constructed above will coincide with
the analytic one~\cite{Vasilescu82} on the polynomials. This gives
\begin{cor}\label{co:holom}
Restriction of the monogenic calculus for a commuting
$n$-tuple $T$ to holomorphic functions produces
the analytic functional calculus.
\end{cor}

\subsection{Spectral mapping theorem}
A good notion of functional calculus should be connected with a good
notion of spectrum via the spectral mapping theorem%
\comment{ and spectral decomposition}.
In the analytic calculus~\cite[II.2.23]{Helemskii93} and
the functional calculus of self-adjoint
operators~\cite[Theorem VII.1(e)]{SimonReed80} the homomorphism property
plays a crucial role in proofs of spectral mapping theorems. Nevertheless,
in our versions these results are also preserved.

It is easy to check that formula~\cite[(3.5)]{Ahlfors86}
can be adapted in the following way:
\begin{lem}
Let $(-c^{-1}d)\not\in \sigma(\vecbf{T})$ and $\vecbf{x}\not=-c^{-
1}d$.
Let $g$ be the action of $\matr{a}{b}{c}{d}$ in $\algbf{A}$. Then
\begin{displaymath}
g(\vecbf{T})-g(\vecbf{x})=(c\vecbf{x}+d)^{*-1}(\vecbf{T}-
\vecbf{x})(c\vecbf{T}+d)^{-1}.
\end{displaymath}
\end{lem}
\begin{thm}[Spectral mapping theorem]
\begin{displaymath}
\sigma(\widehat{f}(\vecbf{T}))=\{\widehat{f}(\vecbf{x}) \such
\vecbf{x}\in\sigma(\vecbf{T})\}.
\end{displaymath}
\end{thm}
\begin{proof}
Let $x\in\sigma(\vecbf{T})$ and $\vecbf{y}=f(\vecbf{x})$ then
\begin{align*} %\begin{eqnarray*}
\widehat{f}(\vecbf{T})-\vecbf{y}&=\widehat{f}(\vecbf{T})-\widehat{f}
(\vecbf{x})\\
  &=\int_V f(g) g(\vecbf{T})\, dg -\int_V f(g) g(\vecbf{x})\, dg\\
  &=\int_V f(g) \left[g(\vecbf{T})-g(\vecbf{x})\right]\, dg \\
  &=\int_V f(g) (c\vecbf{x}+d)^{*-1}(\vecbf{T}-
\vecbf{x})(c\vecbf{T}+d)^{-1}\, dg.
\end{align*} %\end{eqnarray*}
Thus $f(\vecbf{T})-\vecbf{y}$ belongs to the closed proper left ideal
generated by $\vecbf{T}-\vecbf{x}$.

Otherwise let $\widehat{f}(\vecbf{x})\not = \vecbf{y}$ for all $\vecbf{x}
\in \Omega \supset \sigma(\vecbf{T})$. Analogously to Lemma~\ref{le:spectr}
with the help of Theorem~\ref{th:compose} we conclude that the function
\begin{displaymath}
\widehat{[\phi_{(y,1)}f]}(\vecbf{x})=\phi_{(y,1)}\int_V f(g) g(\vecbf{x})\, dg=\int_V f(\phi_{(y,1)}g) g(\vecbf{x})\, dg
\end{displaymath}
is well defined on $\Omega$. Thus we can define an operator
\begin{displaymath}
\widehat{[\phi_{(y,1)}f]}(\vecbf{T})=\int_V f(\phi_{(y,1)}g) g(\vecbf{T})\, dg.
\end{displaymath}
Due to Lemma~\ref{le:spectr} the existence of such an operator
provides us with the invertibility of the operator $\widehat{f}(\vecbf{T}) -\vecbf{y}$.
\end{proof}

\section{Concluding remarks}
\begin{rem}
\textup{It seems worthwhile (if not very natural) to connect the two main
motives
of functional analysis: functional calculus and group representations.
Such a connection causes changes in applications based on the
functional
calculi (quantum mechanics~\cite{Kisil95g}): to quantize one could use not
algebra homomorphisms (observables are usually believed to form an
algebra) but representations of groups (symmetry of the system
under consideration).}
\end{rem}
\begin{rem}
\textup{There is the Cayley transformation~\cite{Ahlfors86} of
the unit ball to the
``upper half-plane'' $x_{n}\geq 0$. This allows one to make a
straightforward modification of the monogenic calculus for
an $n$-tuple of operators where only $T_{n}$ is
semibounded (Hamiltonian!) and the other operators are just unbounded.}
\end{rem}
\begin{rem}\label{re:other}
\textup{The classic
Dunford-Riesz~\cite[Chap.~IX]{RieszNagy55} calculus can be obtained
similarly using the representation of
$\object{SL}{(2,\Space{R}{})}$ by fractional-linear mappings~\cite{MTaylor86}
of the complex
line $\Space{C}{}$. The functional calculus of a self-adjoint
operator~\cite[Chap.~VII]{SimonReed80} can be obtained from
affine transformations of the real line. But a non-formal integral formula
is possible only through an integral representation of the $\delta$-function:
in this way we have arrived, for example, at the Weyl functional
calculus~\cite{Anderson69}.}
\textup{Biholomorphic automorphisms of unit ball in $\Space{C}{n}$
also form a sufficiently large group (subgroup of the M\"obius group) to
construct a function theory~\cite{Kisil95a}. Thus, we can develop a
functional calculus based on the theory of several complex variables.
Due to its biholomorphic
covariance~\cite{CurtoVasil94a,CurtoVasil93a} it must coincide
with the well-known analytic calculus~\cite{Vasilescu82}.}
\end{rem}
\begin{rem}
\textup{Our Definition~\ref{de:calc} is connected with
\emph{Arveson-Connes spectral
theory}~\cite{\comment{Arveson80,Connes73,}Takesaki83}. The main differences
are: (i) most of our groups are non-commutative;
(ii) we use a space of analytic functions (not functions on the group
directly) as a model of functional calculus and definition of a
spectrum.
It seems that the first feature is mainly due to the second
one.}
\end{rem}

\comment{Our technique is mostly
elementary unlike the finest homological approach of J.~L.~Taylor.}
\newcommand{\noopsort}[1]{} \newcommand{\printfirst}[2]{#1}
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  \newcommand{\vrm}{\textup{V}}
\makeatletter \renewcommand{\@biblabel}[1]{\hfill#1.}\makeatother
\begin{thebibliography}{10}

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