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To display a typeset version of a journal * % * article easily, we suggest that you either view the HTML version or * % * retrieve the article in DVI, PostScript, or PDF format. * % ************************************************************************** % Author Package file for use with AMS-LaTeX 1.2 % Date: 28-MAY-1996 % \pagebreak: 0 \newpage: 0 \displaybreak: 0 % \eject: 0 \break: 0 \allowbreak: 0 % \clearpage: 0 \allowdisplaybreaks: 0 % $$: 0 {eqnarray}: 2 % \linebreak: 0 \newline: 0 % \mag: 0 \mbox: 1 \input: 0 % \baselineskip: 0 \rm: 0 \bf: 9 \it: 0 % \hsize: 0 \vsize: 0 % \hoffset: 0 \voffset: 0 % \parindent: 0 \parskip: 0 % \vfil: 0 \vfill: 0 \vskip: 0 % \smallskip: 0 \medskip: 0 \bigskip: 0 % \sl: 0 \def: 1 \let: 0 \renewcommand: 2 % \tolerance: 0 \pretolerance: 0 % \font: 0 \noindent: 0 % ASCII 13 (Control-M Carriage return): 0 % ASCII 10 (Control-J Linefeed): 0 % ASCII 12 (Control-L Formfeed): 0 % ASCII 0 (Control-@): 0 % Special characters: 0 % % %\controldates{16-JUL-1996,16-JUL-1996,16-JUL-1996,23-JUL-1996} \documentclass{era-l} %\issueinfo{2}{1}{}{1996} %\NeedsTeXFormat{LaTeX2e} % \theorembodyfont{\slshape} \newtheorem{thm}{Theorem}[section] \newtheorem{prop}[thm]{Proposition} \newtheorem{lem}[thm]{Lemma} \newtheorem{cor}[thm]{Corollary} % \theorembodyfont{\upshape} \theoremstyle{definition} \newtheorem{defn}[thm]{Definition} \newtheorem{notation}[thm]{Notation} \newtheorem{example}[thm]{Example} \newtheorem{conj}[thm]{Conjecture} \newtheorem{prob}[thm]{Problem} \theoremstyle{remark} \newtheorem{rem}[thm]{Remark} %\providecommand{\AMSMSC}[2]{ %\subjclass{Primary #1; Secondary #2.}} \newcommand{\person}[1]{\textsc{#1}} \hyphenation{di-men-sio-nal} \newcommand{\comment}[1]{} %\def \newcommand\smnegskip{\mskip-0.6\thinmuskip} \usepackage{amsfonts} \newlength{\curmathsize} % % Objects % \newcommand{\algebra}[1]{{{\mathfrak #1}}} %% Change \Cliff, \object to simpler definitions since latex2html %% doesn't yet handle optional arg newcommands properly [mjd,1996/07/30] %%\newcommand{\Cliff}[2][\comment]{{\mathbf{Cl}(#1,#2)}} \newcommand{\Cliff}[1]{{\mathbf{Cl}(#1)}} %%\newcommand{\object}[2][\,]{{\mathrm{#2}#1}} \newcommand{\object}[2]{{#1\mathrm{#2}}} \newcommand{\Space}[2]{{ {{\mathbb #1}^{#2}} }} \newcommand{\such}{\,\mid\,} \newcommand{\FSpace}[2]{{{ #1_{#2} }}} \newcommand{\bos}{{\mathcal B}} % % Unary operations % \newcommand{\Cstar}{$C^{*}$} \newcommand{\norm}[1]{\left\| #1 \right\|} \newcommand{\modulus}[1]{\left|#1\right|} % % Binary operations % \newcommand{\scalar}[2]{\langle #1,#2\rangle} % %\providecommand{\eqref}[1]{\textup{(\ref{#1})}} \newcommand{\unit}{\mathbf{1}} \renewcommand{\sp}[1]{\mathrm{sp}_{#1}\,} \newcommand{\res}[1]{\mathrm{res}_{#1}\,} \newcommand{\matr}[4]{{{ \left( \begin{array}{cc} #1 & #2 \\ #3 & #4 \end{array}\right) }}} %% Not used: %%\newcommand{\spt}{\object{supp}} \newcommand{\vecbf}[1]{\mathbf{#1}} \newcommand{\algbf}[1]{ %\mbox \text{\boldmath$\mathfrak{A}$}} \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} \newcommand{\singleletter}[1]{#1} \newcommand{\switchargs}[2]{#2#1} \newcommand{\irm}{\textup{I}} \newcommand{\iirm}{\textup{II}} \newcommand{\vrm}{\textup{V}} \makeatletter \renewcommand{\@biblabel}[1]{\hfill#1.}\makeatother \begin{thebibliography}{10} \bibitem{Ahlfors86} L.~V. Ahlfors, {\em {M{\"o}bius} transformations in {$\Space{R}{n}$} expressed through $2\times 2$ matrices of {Clifford} numbers}, Complex Variables Theory Appl. %{\bf \textbf{ 5} (1986), no.~2, 215--224. \MR{88a:15052} \bibitem{Anderson69} R.~F.~V. 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