数理解析研究所講究録
1487 巻 2006 年 13-26

13

Geometry of Banach Spaces Predual toCorona Problem

and$H^{\infty}$

Alexander Brudnyi*
Department of Mathematics and Statistics
University of Calgary, Calgary
CanadaAbstract

In this paper we present a new general approach to the multi-dimensional

corona problem for the ball and polydisk. We also describe some propertiesof the Banach spaces predual to the spaces$H^{\infty}$

defined on these domains.

1. Corona Problem
1.1. In the present paper we discuss one of the fundamental problems in the theoryof uniform algebras known as the corona problem. Such a problem was originallyisraised by S. Kakutani in 1941 and asked whether the open unit diskof bounded holomorphic
dense in the maximal ideal space of the algebra
functions on with the supremum norm. This problem was answered affirmatively
by L. Carleson in 1962 [C1]. Let us recall that for a uniform algebra of continuousfunctions defined on a Hausdorff topological space $X$ , the maximal ideal spaceequipped with the weak*-topology
is the set of all nonzero homomorphisms
is a compact, known as the Gelfand topology. Thenof the dual spacebyHausdorff space. Any function $f\in A$ can be considered as a function onmeans of the Celfand transform:$\mathrm{D}\subset \mathbb{C}$$H^{\infty}(\mathrm{D})$$\mathrm{D}$$A$$\mathcal{M}(A)$$Aarrow \mathbb{C}$$A^{*}$$\mathcal{M}(A)$$\mathcal{M}(A)$$\hat{f}(m):=m(f)$,$m\in \mathcal{M}(A)$.

,separates points on $X$ , then $X$ is naturally embedded into
$x\in X$ , whereis the evaluation functional at . In this case we identify $X$ with. The
its image inand bydenote the closure of $X$ inset
is called corona. Then the corona problem is to determine
whether $C(A)=\emptyset$ for certain algebras . As it was mentioned above, the celebrated.
Carleson corona theorem states that $C(A)=\emptyset$ for

If

$A$

$\mathcal{M}(A),$$x$$\delta_{x}$$\mathcal{M}(A)$$xrightarrow\delta_{x}$$\mathcal{M}(A)$$\overline{X}\subset \mathcal{M}(A)$$C(A):=\mathcal{M}(A)\backslash \overline{X}$$A$$A=H^{\infty}(\mathrm{D})$’Research supported in part by NSERC.
.
, Secondary,
2000 Mathematics Subject Classification. Primary, Banach algebra.
Key words and phfases. Corona problem, Banach space predual to$32\mathrm{A}65$$32\mathrm{A}37$$H^{\infty}$

14
Let us briefly describe further developments in the corona problems for Riemannsurfaces.
A number of authors have proved the corona theorem for finite bordered Riemannsurfaces, see, e.g., [JM] for the references. We will single out the proof given by Forelli[F] in which he constructed a linear projector fromonto the subspaceof bounded holomorphic functions invariant with respect to the action of thefundamental group of a bordered Riemann surface satisfying$H^{\infty}(\mathrm{D})$$p$$H_{G}^{\infty}$

$G$

$S$

$p(fg)=p(f)g$ ,$f\in H^{\infty}(\mathrm{D}),$$g\in H_{G}^{\infty}$.

Using this projector one can easily prove the corona theorem for . Further developments of this idea were given by Carleson [C2], Jones and Marshall [JM] andthe author [Brl], [Br2]. In particular, in [JM] the authors constructed a similar
Forelli-type projector for Riemann surfaces such that critical points of Green’s, and using this proved thefunction on form an interpolating sequence forcorona theorem for such . This class of Riemann surfaces contains, e.g., surfaces, whereof the form
is homogeneous. In this way the authorv obtained
another proof of the result of Carleson [C2]. It was also observed in [JM] that every
Riemann surface from the above class is of Widom type, i.e., its topology grows
slowly as measured by the Green function. Finally, by using different techniques,in [GJ] the corona theorem was proved for all Denjoy domains, i.e., domains of the
R.
form
where
There are also examples of (nonplanar) Riemann surfaces for which the corona
theorem fails, see, e.g., references in [JM] and [Brl]. However, the general corona
problem for planar domains is still open, as is the problem in several variables forthe ball and polydisk.
1.2. In this part we consider matrix-valued corona theorems which, in a sense, arenonlinear analogs of the corona theorem. First, let us recall that the corona theorem
for a uniform algebra on a topological space $X$ such that separates points on$X$ is equivalent to the following statement, see, e.g., [G]:, $f_{n}\in A$ satisfying the corona condition
For every collection
$S$

$S$

$S$

$H^{\infty}(S)$$S$$E\subset \mathbb{R}$$\mathbb{C}\backslash E$$\overline{\mathbb{C}}\backslash E$$E\subset$

$A$

$A$

$f_{1},$$\ldots$$\max_{1\leq 1\leq n}.|f_{1}(x)|\geq\delta>0$

there are functions$g_{1},$$\ldots,g_{n}\in A$,

for all$x\in X$,

(1.1)

such that$\sum_{i=1}^{n}f_{1g_{1}}=1$

.

(1.2)

Similarly, the matrix-valued corona problem is formulated as follows:
Let $F=(f_{ij})_{f}1\leq i\leq k,$ $1\leq j\leq n,$ $k<n$ , be a matrix with entries in suchthat the family of all minors of order of satisfies the corona condition (1.1). Is
matrixwith entries in A whose first rows coincide utth andthere an$det(\tilde{F})=1^{q}$such that
Clearly, the necessary condition of the existence of such matrices for all is thevalidity of the corona theorem for . However, it is not sufficient: some topological
do not allow to find such for a generic , see [L].
characteristics of
$A$

$k$

$n\cross n$$F$$\tilde{F}$

$F$

$k$

$n$

$A$

$\mathcal{M}(A)$$\tilde{F}$

$F$

15
defined
We will formulate now some positive results in this area for algebras
on Riemann surfaces.
was proved by Tolokonnikov, see,
The matrix-valued corona theorem fore.g., [Ni]. Later on using the above cited Forelli theorem, the Oka principle forholomorphic vector bundles proved by Grauert [Gr] together with Beurling typehe also proved in [T] the matrix-valued corona theorem fortheorems forbordered Riemann surfaces. Recently we proved, see [Br2], the matrix-valued corona
theorem for subdomains of coverings of bordered Riemann surfaces such thatthe embedding $Darrow C$ induces an injective homomorphism of the correspondingonfundamental groups. In the proof we used a new Forelli-type theorem for
Stein manifolds, see [Brl], a generalized Oka principle for certain complex vector
bundles on maximal ideal spaces of some algebras of bounded holomorphic functions.
on Stein manifolds, see [Br2], together with the classical techniques ofis aRecently we obtained also the following result, see [Br3]. Suppose thatRiemann surface of a finite type (that is, the rank of the first cohomology groupis finite). Let be a Galois covering of . Then the matrix-valued corona. Surprisingly, together with some techniques developed
theorem is valid forvanishing theorems.
in [Brl] and [Br2] we used in the proof some
Also, we conjecture that the matrix-valued corona theorem is valid for any (notnecessarily Galois) covering of a Riemann surface of a finite type.
$H^{\infty}$$H^{\infty}(\mathrm{D})$$H^{\infty}(\mathrm{D})$

$D$

$C$

$H^{\infty}$$H^{\infty}(\mathrm{D})$

$S$

$S$

$C$

$H^{1}(S, \mathbb{Z})$$H^{\infty}(C)$$L_{2}\mathrm{K}\mathrm{o}\mathrm{d}\mathrm{a}\mathrm{i}\mathrm{r}\mathrm{a}_{r}\mathrm{N}\mathrm{a}\mathrm{k}\mathrm{a}\mathrm{n}\mathrm{o}$2. Banach Spaces Predual to$H^{\infty}$.
2.1. Our purpose is to present a general approach to the corona problems for.
spacestopredualTo this end we first describe the structure of Banachseparates
Suppose that $M$ is a complex manifold such that the algebra
points on $M$ . Then there is a complex Banach space $X(M)$ such that $X(M)^{*}=$. Such a space is constructed as follows: it is the minimal closed subspace$m\in M$ . It isof the dual spacecontaining all evaluation functionalsof the set
easy to see that the closed unit ball of $X(M)$ is the closure in$H^{\infty}$$H^{\infty}$$H^{\infty}(M)$$H^{\infty}(M)$$H^{\infty}(M)^{*}$$\delta_{m},$$H^{\infty}(M)^{*}$$\{\sum_{i=1}^{k}\mathrm{q}_{n_{2}}\delta_{m_{i}}$:$\mathrm{q}\in \mathbb{C}$, $m_{i}\in M$ ,$k\in \mathrm{N},$$\sum_{i=1}^{k}|\mathrm{c}_{i}|\leq 1\}$.

, is injective and: $Marrow X(M),$
It is also obvious that the map
holomorphic, that is, for every $f\in H^{\infty}(M)(=X(M)^{*})$ the function. In what follows we identify $M$ with its image $\phi_{M}(M)\subset X(M)$ .
$\phi_{M}$$m-\rangle\delta_{m}$$f\mathrm{o}\phi_{M}\in$$H^{\infty}(M)$consists of linearly independent
Proposition 2.1 The familyelements and the space generated by this famdy is dense in $X(M)$ .
$\{\delta_{m}\}_{m\in M}\subset X(M)$

Sketch of the Proof. The first statements follows from the fact that any finite set. The second statement
an interpolating sequence forof points in
is obvious.
$H^{\infty}(M)$$M\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{s}\square$

16

Let $P_{2}^{h}[X(M)]$ be the linear space of holomorphic polynomials on $X(M)$ generated by polynomials of the form$p(v):=f(v)g(v)$ ,$f,g\in H^{\infty}(M)$.

By definition each $f\in H^{\infty}(M)$ admits a continuous extension to$f(\xi):=\xi(f)$,$f\in H^{\infty}(M),$$\xi\in H^{\infty}(M)^{*}$$H^{\infty}(M)^{*}$

given by.

Thus every polynomial from $P_{2}^{h}[X(M)]$ also admits an extension to. (Without loss of generality we assume that $P_{2}^{h}[X(M)]$ is already defined on
Let 1 be the functional on
corresponding to the constant function 1. By$Z(M)$ we denote the affine subspace ofdefined by$H^{\infty}(M)^{*}$$H^{\infty}(M)^{*}.)$$H^{\infty}(M)^{*}$$H^{\infty}(M)^{*}$$Z(M):=\{\xi\in H^{\infty}(M)^{*} :1(\xi)=1\}$.

(Observe that $M\subset \mathcal{Z}(M)\cap X(M)$ and the closure in the weak’-topology on$H^{\infty}(M)^{*}$

of $Z(M)\cap X(M)$ coincides with$\mathcal{Z}(M).)$

Finally, for a bounded subset
by $h_{2}(S)$ we denote the polynomiallyconvex hull of with respect to polynomials from $P_{2}^{h}[X(M)]$ , that is,$S\subset H^{\infty}(M)^{*}$$S$$\xi\in h_{2}(S)$for all$\Leftrightarrow$$|p( \xi)|\leq\sup_{\mu\in S}|p(\mu)|$$p\in P_{2}^{h}[X(M)]$.

This definition is well-defined because every $p\in P_{2}^{h}[X(M)]$ is bounded on a boundedby the Banach-Alaoglu theorem. The one can easily check
subset of$H^{\infty}(M)^{*}$

Proposition 2.2 The $maximal$ ideal space
the set $Z(M)\cap h_{2}(M)$ .

$\mathcal{M}(H^{\infty}(M))$

Let us consider another characterization ofnorm on $P_{2}^{h}[X(M)]$ by the formula$||p||:= \sup_{v\in B}|p(v)|$,of$\mathcal{M}(H^{\infty}(M))$$p\in P_{2}^{h}[X(M)]$coincides with$H^{\infty}(M)$. Let us introduce the,where is the unit closed ball in $X(M)$ . Let $X_{2}(M)$ be the minimal closed subspaceof the dual space $(P_{2}^{h}[X(M)])^{*}$ (constructed with respect to this norm) generated$v\in X(M)$ . One can prove using the Hahn-Banach theorem
by delta functionals
that the closed unit ball of $X_{2}(M)$ is the closure in $(P_{2}^{h}[X(M)])^{*}$ of the set$B$$\delta_{v},$$\{\sum_{i=1}^{k}\mathrm{c}_{v}\dot{.}\delta_{v\iota}$:$c_{i}\in \mathbb{C},$$v:\in B,$$k\in \mathrm{N},$$\sum_{:=1}^{k}|c_{i}|\leq 1\}$.

$v\in$:
We identify $X(M)$ with its image in $X_{2}(M)$ under the map$X(M)$ . Then clearly every element of
belongs to the dual space
there. Using the Banach-Alaoglu theorem one can also checkand has normin the weak’-topology. We will call elementsthat $P_{2}^{h}[X(M)]$ is dense inhomogeneous polynomials of degree 2 on $X(M)$ . Let be the minimalof$\phi_{X(M)}$$\leq 1$

$X_{2}(M)^{*}$$X_{2}(M)^{*}$

closed subspace of $X_{2}(M)$ containing $M(=\phi_{X(M)}(M))$ .

$v\vdasharrow\delta_{v},$$X_{2}(M)^{*}$$\mathcal{P}_{2}^{h}[X(M)]$

$V$

17

Proposition 2.3$H^{\infty}(M)$

.

(a)

$V$

is isometric to $X(M)$ and $X_{2}(M)^{*}|_{V}$ is isometric to$TheclosureofVintheweak^{*}- topologyof(P_{2}^{h}[X(M)])^{*}isisomet\dot{m}ctoH^{\infty}(M)^{*}$ .

(b)

Proof. (a) Clearly there exists a continuous linear map from $X(M)$ into . By
we denote the closure of the image under this map of the unit ball in $X(M)$ . Let$B(V)$ be the closed unit ball in. Assume that is. By definition,$B(V)$
a proper subset of. Then by the Hahn-Banach theorem there is an element$f\in X_{2}(M)^{*}$ such that $f(v_{0})=1$ for some $v_{0}\in B(V)$ , but $\sup_{w\in\hat{B}}|f(w)|=s<1$ .
Let $f’=f|_{M}$ . Then the element $h:=f’\cdot 1\in P_{2}^{h}[X(M)]$ and $h-f$ equals on$M$ . Thus
$h=f$ on . But by definition $||h||_{V}=s<1$ , a contradiction. Hence,$\hat{B}=B(V)$ and $X(M)$ is isometric to. The second part of (a) is then obvious.
(b) By definition, for every $f\in V^{*}$ there is an element $\hat{f}\in P_{2}^{h}[X(M)]\subset X_{2}(M)^{*}$
such that. This gives a linear continuous projection
and$P_{2}^{h}[X(M)]arrow V^{*}$ . Its dual map is an embedding: $V^{**}arrow(P_{2}^{h}[X(M)])^{*}$ continuous
in the corresponding weak“-topologies and such that the restriction of to
co$Varrow(P_{2}^{h}[X(M)])^{*}$. This implies the required
incides with the original embedding
statement.
Identifying with $X(M)$ we have the following decomposition of $X_{2}(M)^{*}$ .
Let $f\in X_{2}(M)^{*}$ . By definition, $g:=f|_{X(M)}\in H^{\infty}(M)$ . Thus
$V$$\hat{B}\subset B(V)$$V$$\hat{B}$$\hat{B}$

$0$

$V$

$V$

$\hat{f}|_{V}=f$$||\hat{f}||_{X_{2}(M)}=||f||_{V}$

$i$

$i$

$V$

$\square$$V$$f=g+(f-g)$ ,$g\in H^{\infty}(M),$$(f-g)\in X(M)^{\perp}$,$||g||\leq||f||$,$||f-g||\leq 2||f||$.

is the norm on $X_{2}(M)^{*}$ . This implies the natural decomposition $X_{2}(M)$ $:=$
where $X(M)^{0}\subset X_{2}(M)$ is such that $(X(M)^{0})^{\perp}=H^{\infty}(M)$ . Using: $X(M)arrow X_{2}(M)$ with the graphthis we identify the image of the map$X(M)arrow
X(M)^{0}$: $X_{2}(M)arrow X(M)^{0}$ ismapof the:where
the natural linear projection. It is easy to check thatis a complex Banach
submanifold of $X_{2}(M)$ biholomorphic to $X(M)$ . Let : $X_{2}(M)arrow X(M)$ be thenatural projection. For any $p\in P_{2}^{h}[X(M)]$ by
we denote the linear
functional representing . It is easy to proveHere$||\cdot||$$X(M)\oplus X(M)^{0}$$\phi_{X(M)}$$\pi_{0}0\phi_{X(M)}$$\Gamma_{X(M)}$$\pi_{0}$$\Gamma_{X(M)}$$\pi$$\hat{p}\in X_{2}(M)^{*}$$p$

Proposition 2.4
is the set of zeros
on $X_{2}(M)^{*}$ .
polynomials
2
degreeofof$\Gamma_{X(M)}$$\square of$the family$\{p\mathrm{o}\pi-\hat{p} :p\in P_{2}^{h}[X(M)]\}$

Let
is a complex
denote $\phi_{X(M)}(Z(M)\cap X(M))$ . Then
$Z(M)\cap
X(M)$
. By
biholomorphic to
we denote thesubmanifold of
in the weak*-topology of $(P_{2}^{h}[X(M)])^{*}$ . (Recall that according toclosure of
our construction $X_{2}(M)$ is a closed subspace of $(P_{2}^{h}[X(M)])^{*}.)$ Then we have$\tilde{\mathcal{Z}}(M)$$\tilde{\mathcal{Z}}(M)\subset\Gamma_{X(M)}$$\mathrm{c}1(\tilde{Z}(M))$$\Gamma_{X(M)}$$\tilde{Z}(M)$Proposition 2.5$\mathcal{M}(H^{\infty}(M))=\mathrm{c}1(\tilde{Z}(M))\cap \mathcal{Z}(M)$.

. Then $\xi\in X(M)^{**}\subset$
Sketch of the Proof. Assume that$p(\pi(\xi))-\hat{p}(\xi)=0$
by Proposition 2.3, andfor all $p\in P_{2}^{h}[X(M)]$ (here
we naturally extend
and to $(P_{2}^{h}[X(M)])^{*}8S$ continuous in the weak’-topology$4\in \mathcal{M}(H^{\infty}(M))$$(P_{2}^{h}[X(M)])^{*}$$p\mathrm{o}\pi$$\hat{p}$18functions). Also, by definition, $\xi\in Z(M)$ . Let
be a bounded netthat converges in the weak*-topology to $\pi(\xi)\subset X(M)^{**}$ . Passing if necessary toa subnet we may assume without loss of generality thatconverges tosome Xin the weak’-topology of $X_{2}(M)^{**}$ so that $\pi(\xi)=\pi(\chi)$ . From
here for every $p\in P_{2}^{h}[X(M)]$ we have
. Since the weak’-topology on$X_{2}(M)^{**}$ is generated by elements, we have. The converse
implication of the theorem is obvious.
$f_{n}\in H^{\infty}(M)$ be such that
2.2. Let$\{\chi_{\alpha}\}\subset X(M)$$\{\phi_{X(M)}(\chi_{\alpha})\}$$\in \mathrm{c}1(\overline{Z}(M))$$\hat{p}(\xi)=\hat{p}(\chi)$$\xi=\chi\in \mathrm{c}1(\tilde{Z}(M))$$\hat{p}$$\square$$f_{1},$$\ldots,$$1\leq|.\leq n\mathrm{m}\mathrm{a}\mathrm{x}||f_{i}||_{H}\infty(M)\leq 1$

and$\max_{1\leq i\leq n}|f_{1}(x)|\geq\delta>0$

By $K(f_{1}, \ldots, f_{n})$ we denote the set of zeros offrom Theorem 2.5 follows easily$f_{1},$

Proposition 2.6 The corona theorem is valid for$K(f_{1}, \ldots, f_{n})$ as above one has$\ldots,$,

for all.

in $X(M)\cap Z(M)$ . Then$f_{n}$$H^{\infty}(M)$$\mathrm{c}1(\phi_{X(M)}(K(f_{1}, \ldots, f_{n})))\cap Z(M)=\emptyset$$x\in M$.

if and only if for every$\square$Assume now that $M$ stands for the open unit Euclidean ballor the open unit
. The first step in the proof of the corona theorem for such $M$ isinpolydisk
the following result whose proof is based on Theorem 3.4 of the next section.
$\mathrm{B}^{N}$$\mathrm{D}^{N}$$\mathbb{C}^{N}$

Theorem 2.7 There is a constant $c=c(N)$ such that$||\phi_{X(M)}(v)-v||_{X_{2}(M)}\geq c\cdot(dist(v,M))^{2}$Here
dist $(v,M):= \inf_{z\in M}||v-z||_{X(M)}$ ,for all$v\in X(M)$.

$v\in X(M)$ .

In particular, this theorem implies thatdist$(\phi_{X(M)}(K(f_{1}, \ldots, f_{n})), K(f_{1}, \ldots, f_{n}))\geq c\delta^{2}$.

However, we still don’t know whether this inequality leads to the required conclusionof Proposition 2.6. Let us mention that taking into account the latter inequality
one can consider the statement of the corona theorem as a nonlinear analog of theisHahn-Banach separation theorem. Namely, if we assume for a while that$\mathrm{c}1(\phi_{X(M)}(K(f_{1},\ldots,f_{n})))\cap
Z(M)=\emptyset$
a linear map, then the required statementin ourfollows from the classical Hahn-Banach theorem. But, unfortunately,
map....
case is a holomorphic quadratic
Finally, we will mention that from Theorem 2.7 it follows$\phi_{X(M)}$$\phi_{X(M)}$

Proposition 2.8 There is a continuous plunsubharmonic function of logarithmic
growth on $X(M)equals-\infty$ on $M$ such that for some constant $c\sim=c(\sim N)$$g$$g(v)\geq\sim c+\ln(dist(v,M))$

for all $v\in X(M)$ .

19

Let us consider now the uniform algebra of functions on the closed unit ballbe the maximal
and the function . Let
generated byideal space of this algebra.
$B\subset X(M)$$H^{\infty}(M)$Problem 2.9 Is it true that$B$$e^{g}$$\mathcal{M}(B)$is dense in the Gelfand topology of$\mathcal{M}(B)^{q}$

The positive answer in this problem leads to the solution of the corona problem foras abovewith. Indeed, the collection of functions
andsatisfy condition (1.1) (i.e., the corona condition) on . Then from the positiveinanswer in Problem 2.9 it follows that there are complex polynomials
vuch thatvariables$\mathrm{D}^{N}$$\mathrm{B}^{N}$$e^{g},$$f_{1},$$\ldots,$$f_{1},$$f_{n}$$f_{n}$$\ldots,$$B$$p_{0},$$e^{\mathit{9}},$$f_{1},$$\ldots,$$\ldots p_{n}$$f_{n}$

for all$|p_{0}(v)e^{g(v)}+ \sum_{k=1}^{n}p_{k}(v)f_{k}(v)|>\frac{1}{2}$$v\in B$.

to $M$ is a holomorphic polynomial
Since $M\subset B$ and the restriction of each
only, $0\leq k\leq n$ , the latter inequality implies the solution of the.
,corona problem for the family
a closed ball in a
Let us consider the general version of Problem 2.9 forfunctionplurisubharmonic
Banach space and for an arbitrary logarithmicallyon such that is of logarithmical growth. Then the only result obtained for this$p_{k}$

in$f_{1}|_{M},$ $\ldots f_{n}|_{M}$$f_{1},$$\ldots$$f_{n}\in H^{\infty}(M)$

$B$

$V$

$V$

$e^{g}$$g$general setting is the following
Proposition 2.10 Suppose that the function.
in the Gelfand topology of$e^{g}$

is a norm on V. Then $B$ is dense$\mathcal{M}(B)$Remark 2.11 Of course, Problem 2.9 is also valid for the trivial case of

3. Geometry of Banach Spaces Predual to$e^{g}\equiv\omega nst$.

$H^{\infty}$

3.1. Let $M$ be a Caratheodory hyperbolic complex manifold of bounded geometry,
i.e., there exists a constant $C\geq 1$ such that every point $z\in M$ has a neighbourhood
admitting a biholomorphism$U_{z}$$\psi_{z}$:$U_{z}arrow \mathrm{B}^{N}$

such that$\frac{1}{C}d_{E}(\psi_{z}(v_{1}),\psi_{z}(v_{2}))\leq d_{M}(v_{1},v_{2})\leq Cd_{E}(\psi_{z}(v_{1}),\psi_{z}(v_{2}))$for all$v_{1},$$v_{2}\in U_{z}$.

is the Caratheodory distance onis the Euclidean distance on
andHere$M$ . Let. This means that for every
be a separated -net with respect to$s_{2}\in S$ ,$z\in M$ there exists $s\in S$ such that $d_{M}(z, s)<\epsilon$ and for every distinct. Then from the definition of the bounded geometry for a sufficiently
small (depending on ) it follows that$\mathbb{C}^{N}$$d_{E}$$S_{\epsilon}\subset M$$\epsilon$$d_{M}$$d_{M}$$s_{1},$$d_{M}(s_{1}, s_{2})\geq\epsilon$$C$$\epsilon$

Proposition 3.1 The restriction
determines an isometricembedding of
intocontinuous in the corresponding weakd-topologies.
$H^{\infty}(M)arrow H^{\infty}(M)|_{S_{\mathrm{e}}}$$H^{\infty}(M)$$l^{\infty}(S_{\epsilon})$20

Thus this embedding determines a surjective bounded linear map li $(S_{\epsilon})arrow X(M)$of norm 1 whose dual map coincides with the embedding of the proposition. The
go under this map to the corresponding-functionals of points ofinfunctionals of points of in $X(M)$ . In particular, we obtain$\delta$$l^{1}(S_{\epsilon})$$S_{\epsilon}$$\delta-$$S_{\epsilon}$

Proposition 3.2 Given $v\in X(M)$ andsuch that$v= \sum_{t\in S_{\mathrm{e}}}\alpha_{t}\delta_{t}$withthere is a sequence$\epsilon>0$$||v||_{X(M)} \leq\sum_{t\in S}.|\alpha_{t}|\leq||v||_{X(M)}+\epsilon$$\{\alpha_{t}\}_{t\in S_{\epsilon}}\subset \mathbb{C}$.

Remark 3.3 (1) An open and interesting question is whether $X(M)$ possesses a
Shauder basis.
(2) A similar result can be proved for any $X(M)$ with $M$ Caratheodory hyperbolic,where instead of one chooses “more dense“ countable subsets of M. (We don’tknow whether it is possible to take here separated -nets, as well.)$S_{\epsilon}$$\epsilon$

3.2. In this subsection $M$ stands for. Let $B(M)$ be the group of bior$M$holomorphic transformations of . Then each $g\in B(M)$ determines an isometry$H^{\infty}(M)arrow H^{\infty}(M),$ $g^{*}f:=f\circ g$ , continuous in the weak’-topology of.
This implies existence of a linear isometry : $X(M)arrow X(M)$ of the predual space. Identifying $M$ with a subset of $X(M)$ as in the preceding secsuch that
. The group of all such will be also denoted by $B(M)$ .
tion, we have
. By the same symbols we denoteLetbe holomorphic coordinates on
the corresponding elements of. Let $Z(c)\subset X(M),$,be the set of zeros of the equations $z_{1}=c_{1},$, $z_{N}=c_{N}$ . It is a complex affine
subspace of $X(M)$ of codimension $N$ . In the following result 1 is considered as thelinear functional on $X(M)$ corresponding to $1\in H^{\infty}(M)$ .
$\mathrm{B}^{N}$$\mathrm{D}^{N}$$H^{\infty}(M)$$g^{*}:$$\overline{g}$$\overline{g}^{*}=g^{*}$$\overline{g}|_{M}=g$$\overline{g}$$\mathbb{C}^{N}$$z_{1},$$\ldots,$$z_{N}$$H^{\infty}(M)$$c=(c_{1}, \ldots , c_{N})\in \mathbb{C}^{N}$$\ldots$

Theorem 3.4 Suppose that $v\in X(M)$ and $1(v)\neq 0$ . Then for every $c\in 1(v)\cdot M$.
such that
there exits$\overline{g}\in B(M)$$\overline{g}(v)\in Z(c)$This result is the main point in the proof of Theorem 2.7.
Sketch of the Proof. We will show how to prove the result for
proof forfollows easily from this case.
there is a biholomorphism
Recall that for every point
satisfying$M=\mathrm{B}^{N}$. The$\mathrm{D}^{N}$$a\in \mathrm{B}^{N}$(1)$\phi_{a}(0)=a$,$\phi_{a}(a)=0$(2)$\phi_{a}=\phi_{a}^{-1}$(3)$\phi_{a}$has the unique fixed point.

;;$\phi_{a}$:$\mathrm{B}^{N}arrow \mathrm{B}^{N}$

21

This map can be written explicitly by.

$-Z$$a=0$ ,,$\phi_{a}(z)=$$\frac{a-P_{a}z(1-|a|^{2})^{1/2}Q_{a}z}{1<z,a>}=$,$a\in \mathrm{B}^{N}\backslash \{0\}$where$P_{a}z= \frac{<z,a>}{<a,a>}a$,$Q_{a}z=z- \frac{<z,a>}{<a,a>}a$.
the inner product on
and $<z,$, and in this, the boundary of
is defined also for
Observe that
. Using Proposition 3.2. Suppose now that $v\in X(M)$ and l(v)case
. Consider the linear map
withwe present as$z:=(z_{1}, \ldots, z_{n})$ :. Then$\mathbb{C}^{N}$$a>\mathrm{i}\mathrm{s}$$a\in \mathrm{S}^{N}$$\phi_{a}$$\mathrm{B}^{N}$$\neq 0$$\phi_{a}\equiv a$$v$$\Sigma_{t\in S_{\epsilon}}|\alpha_{t}|\leq 2||v||_{X(M)}$$\Sigma_{t\in S_{e}}\alpha_{t}\delta_{t}$$X(\mathrm{B}^{N})arrow \mathbb{C}^{N}$.

$z( \overline{\phi}_{a}(v)):=\sum_{t\in S_{e}}\alpha_{t}z(\phi_{a}(t))$

andAccording to the above condition this series converges uniformly for
we have:
is a continuous map. Also, fortherefore$a\in\overline{\mathrm{B}}^{N}$$\overline{\mathrm{B}}^{N}arrow \mathbb{C}^{N}$$a\in \mathrm{S}^{N}$$z\mathrm{o}\overline{\phi}.(v)$.

$z( \overline{\phi}_{a}(v))=\sum_{t\in S_{\epsilon}}\alpha_{t}a=1(v)\cdot a$:
is identity on the boundary, by the Brower
Since $1(v)\neq 0$ , and)Therefore for every
contains 1(v)fixed point theorem the image of.
for somesuch that
there exists
3.3. In this part we describe in more details the structure of the spaceas an abelian semigroupof. We will consider the
predual to$i=1,2$, bywith multiplication defined for$\overline{\mathrm{B}}^{N}arrow \mathbb{C}^{N}$$+^{z\circ\phi v}1^{-}v$$\cdot\overline{\mathrm{B}}^{N}$$z\mathrm{o}\overline{\phi}.(v)$$c\in 1(v)\cdot \mathrm{B}^{N}$$a\in \mathrm{B}^{N}$$\overline{\phi}_{a}$$\overline{\phi}_{a}(v)\in Z(c)$$\square$$X(\mathrm{D}^{N})$$\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}\overline{\mathrm{D}}^{N}$$\mathrm{D}^{N}$$H^{\infty}(\mathrm{D}^{N})$$z^{i}:=(z_{1}^{i}, \ldots, z_{N}^{i})\in\overline{\mathrm{D}}^{N},$$z^{1}\cdot z^{2}:=$

Next, for every$\alpha\in\overline{\mathrm{D}}^{N}$by$K_{\alpha}$$(z_{1}^{1}\cdot z_{1}^{2}, \ldots, z_{N}^{1} .

z_{N}^{2})$.

we denote the map defined by$K_{\alpha}(z):=\alpha\cdot z$,$z\in \mathrm{D}^{N}$.

, is a linear operator continuous in:
Then. Hence there is a linear operator
the weak’-topology of. We will denote it by theonwhich coincides withpredual tois an abelian semigroup isomorphic
same symbol . Also, the set$K_{\alpha}^{*}$$H^{\infty}(\mathrm{D}^{N})arrow H^{\infty}(\mathrm{D}^{N}),$$K_{\alpha}^{*}f:=f\mathrm{o}$$K_{\alpha}$$X(\mathrm{D}^{N})arrow X(\mathrm{D}^{N})$$H^{\infty}(\mathrm{D}^{N})$$K_{\alpha}$$K_{\alpha}^{*}$$K_{\alpha}$

to
with

$\overline{\mathrm{D}}^{N}$$K:=\{K_{\alpha}\}_{\alpha\in\overline{\mathrm{D}}^{N}}$

Observe, that every
with, the \S ilov boundary of
convergent to$K_{\alpha}$$\alpha\in \mathrm{T}^{N}$$\{\alpha_{n}\}_{1\leq n<\infty}\subset\overline{\mathrm{D}}^{N}$$\mathrm{D}^{N}\subset X(\mathrm{D}^{N})$is a compact operator and every, is an isometry. Moreover, for a sequence
we have$\alpha\in \mathrm{D}^{N}$$\mathrm{D}^{N}$$\alpha\in\overline{\mathrm{D}}^{N}$$\lim_{narrow\infty}K_{\alpha_{\hslash}}(v)=K_{\alpha}(v)$for every$K_{\alpha}$$v\in X(\mathrm{D}^{N})$

.

(3.1)

22

(For

$\alpha\in \mathrm{T}^{N}$, however,For every
action of the group(for $z=(z_{1},$$\mathrm{T}_{v}$does not converge toin the operator norm.)by
we will denote the orbit of with respect to the. Let us determine the Cauchy integral operator on) by the formula$\{K_{\alpha_{n}}\}$$v\in X(\mathrm{D}^{N})$$K_{\alpha}$$v$$\mathrm{T}_{v}$$\{K_{\alpha}\}_{\alpha\in \mathrm{F}^{N}},$$z_{N})\in \mathrm{D}^{N},$$\ldots,$$\xi=(\xi_{1},$$\xi_{N})\in \mathrm{T}^{N}$$\ldots,$$\mathcal{K}_{z}(v):=(\frac{1}{2\pi i})^{N}\int\cdots\int_{\mathrm{T}^{N}}\frac{K_{\xi\iota.\cdot\cdot.\cdot\xi_{N}}(v)}{(\xi_{1}-z_{1})\cdot(\xi_{N}-z_{N})}d\xi_{1}\cdots d\xi_{N}$Proposition 3.5 For every$v\in X(\mathrm{D}^{N})$.

Sketch of the Proof. The statement is obvious for every
according to Proposition 3.2 we have,

$v:=\delta_{t},$$||v||_{X(\emptyset^{N})}- \epsilon\leq\sum_{t\in S_{\epsilon}}|\alpha_{t}|$$\mathcal{K}_{z}(v)=\sum_{t\in S}.\alpha_{t}\mathcal{K}_{z}(\delta_{t})=\sum_{t\in S_{\epsilon}}\alpha_{t}K_{z}(\delta_{t})=K_{z}(v)$$\overline{\mathrm{D}}^{N}arrow X(\mathrm{D}^{N})$$\psi_{v}$the correspondence
holomorphic on.

$v\in X(\mathrm{D}^{N})$Sketch of the Proof. For every(3.2) we have$\psi_{v}$$f\in H^{\infty}(\mathrm{D}^{N})$is holomorphic on

. Now

.

$\square$determines a$zrightarrow \mathcal{K}_{z}(v)$$\mathrm{D}^{N}$by Propositions 3.2, 3.5 and formula$f( \mathcal{K}_{z}(v))=\sum_{t\in S}‘\alpha_{t}f(zt)\in H^{\infty}(\mathrm{D}^{N})$This shows that
(3.1).
Let$t\in \mathrm{D}^{N}$.

Hence,

Proposition 3.6 Given:
continuous map(3.2)

we have$\mathcal{K}_{z}(v)=K_{z}(v)$$v= \sum_{t\in S_{e}}\alpha_{t}\delta_{t}$.

$\mathrm{D}^{N}$.

. The continuity follows easily from formula$\square$$z^{:}=(z_{1}, \ldots, z_{i-1},0, z_{1+1}, \ldots, z_{N})\in\overline{\mathrm{D}}^{N}$. We study the image of$K_{z^{l}}(X(\mathrm{D}^{N}))$.

, be the natural projection. Then as before this map
Let :gives rise to a bounded linear map
which coincides with : on
. (We denote this map also by ) Moreover,is isometric:andis a linear continuous projection. Set
to.
. Clearly we have (for)$\mathrm{D}^{N}arrow \mathrm{D}^{N-1},$$\pi_{\iota’}$$\pi_{i}(z)=z^{i}$$X(\mathrm{D}^{N})arrow X(\mathrm{D}^{N})$$\mathrm{D}^{N}\subset X(\mathrm{D}^{N})$$X(\mathrm{D}^{N-1})$$X(\mathrm{D}_{1}^{N}$$\pi_{i}$$\pi$$\pi_{i}(X(\mathrm{D}^{N}))$$\pi_{i}.$$X(\mathrm{D}^{N})arrow\pi_{i}(X(\mathrm{D}^{N}))$$0\in \mathbb{C}^{N}$$):=\pi_{i}(X(\mathrm{D}^{N}))$

Proposition 3.7$K_{z}:(X(\mathrm{D}^{N}))\subset X(\mathrm{D}_{i}^{N})$

Notice thatand$\bigcap_{i=1}^{N}X(\mathrm{D}_{1}^{N})=X_{0}:=\{c\delta_{0} :$(\pi_{N}\circ\cdots\circ\pi_{1})(X(\mathrm{D}^{N}))=X_{0}$Proposition 3.8 Suppose that for somethat
contains the other.

and$(\pi_{N}\circ\cdots\circ\pi_{1})(v)=1(v)\delta_{0}$there are
. Then one of the sets

$x,y\in\overline{\mathrm{D}}^{N}$$K_{x}(v)=K_{y}(w) \not\in X(\mathrm{D}^{N})\backslash (\bigcup_{1\leq i\leq N}X(\mathrm{D}_{i}^{N}))$c\in \mathbb{C}\}$ロ.

$v,$ $w\in X(\mathrm{D}^{N})$$\psi_{v}(\overline{\mathrm{D}}^{N}),$such$\psi_{w}(\overline{\mathrm{D}}^{N})$23

Sketch of the Proof. From the assumption of the proposition we obtain that
. Without loss of, and $\psi_{v}(x\cdot z)=\psi_{w}(y\cdot z),$. Then since by thegenerality we may assume, e.g., that $y=\alpha\cdot x$ withfor all, we getis an open polydisk in$x,$$y\in(\overline{\mathrm{D}}^{*})^{N},$$z\in \mathrm{D}^{N}$$\overline{\mathrm{D}}^{*}:=\overline{\mathrm{D}}\backslash \{0\}$$\alpha\in\overline{\mathrm{D}}^{N}$$z\in\overline{\mathrm{D}}^{N}$$\psi_{v}(z)=\psi_{w}(\alpha\cdot z)$$\mathrm{D}^{N}$$\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{s}x\cdot \mathrm{D}^{N}\square$

Observe also that for$v \not\in\bigcup_{1\leq i\leq N}X(\mathrm{D}_{i}^{N})$we have$\psi_{v}((\overline{\mathrm{D}}^{*})^{N})=\psi_{v}(\overline{\mathrm{D}}^{N})\backslash \bigcup_{1\leq i\leq N}X(\mathrm{D}_{i}^{N})$For every
sets

(3.3)

we denote the union of all possiblebycontaining . Then from (3.3) and Proposition 3.8 we get$v \in X(\mathrm{D}^{N})\backslash (\bigcup_{1\leq i\leq N}X(\mathrm{D}_{1}^{N}))$$\psi_{w}((\mathrm{D})^{N}\neg)$.

$G_{v}$$v$$\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}3.9G_{v}.\square X(\mathrm{D}^{N})\backslash (\bigcup_{1\leq i\leq N}X(\mathrm{D}_{i}^{N}))$

Remark 3.10 The closure
morphic map St; hereeach
is either or or C.

$\partial_{v}\subset X(\mathrm{D}^{N})$coincides with the image of a holoof, whereis of the form
$G_{v}$$\Omega\subset \mathbb{C}^{N}$$arrow X(\mathrm{D}^{N})$$\Omega=\Omega_{1}\cross\cdots\Omega_{N}$$\overline{\mathrm{D}}$$\mathrm{D}$$\Omega_{i}\subset \mathbb{C}$is the disjoint union of a family of sets:
the map
An interesting question is for whichmap
theis an embedding? (Observe that for:is always not injective.) Let us formulate a partial answer to$v \in X(\mathrm{D}^{N})\backslash (\bigcup_{1\leq i\leq N}X(\mathrm{D}_{i}^{N}))$$v \in\bigcup_{1\leq i\leq N}X(\mathrm{D}_{i}^{N})$$(\overline{\mathrm{D}}^{*})^{N}arrow X(\mathrm{D}^{N})$$\psi_{v}$$\psi_{v}$$\overline{\mathrm{D}}^{N}arrow X(\mathrm{D}^{N})$this question.

Proposition 3.11 Suppose thatas$\Sigma_{k=1}^{l}\mathrm{c}_{k}\delta_{x_{k}},$$v \in X(\mathrm{D}^{N})\backslash (\bigcup_{1\leq \mathfrak{i}\leq N}X(\mathrm{D}_{i}^{N}))$$c_{k}\in \mathbb{C}_{f}x_{k}\in \mathrm{D}^{N}$$V_{1}:=\{v\in X(\mathrm{D}^{N}) :is either presented, or does not belong to the union of the spaces:. Then the mapz_{i}(v)=0\},$ $1\leq i\leq N$$(\overline{\mathrm{D}}^{*})^{N}arrow X(\mathrm{D}^{N})$$\psi_{v}$is an embedding.

Sketch of the Proof. In the first case the proof follows from the fact that any. In the second case we
finite subset ofis an interpolating sequence for. Then it is easy to see
consider the linear map $z:=(z_{1}, \ldots, z_{N})$ :thatis one-t.
:
into a Banach algebra over (withFinally, we will show how to convert
out unit). To this end we introduce multiplication :by the formula$\mathrm{D}^{N}$$H^{\infty}(\mathrm{D}^{N})$$X(\mathrm{D}^{N})arrow \mathbb{C}^{N}$$\overline{\mathrm{D}}^{N}arrow \mathbb{C}^{N}$$z\circ\psi_{v}$$\square$$(\succ \mathrm{o}\mathrm{n}\mathrm{e}$$X(\mathrm{D}^{N})$$\mathbb{C}$$X(\mathrm{D}^{N})\cross X(\mathrm{D}^{N})arrow X(\mathrm{D}^{N})$$v \cdot w:=(\sum_{t\in \mathit{3}_{*}}\alpha_{t}\delta_{t})(\sum_{\epsilon\in S_{\epsilon}}\beta_{\epsilon}\delta_{\delta})=\sum_{t,s\in S_{e}}\alpha_{t}\beta_{\epsilon}\delta_{t\cdot\epsilon}$

.

(3.4)

and
are some presentations of and as
Herein Proposition 3.2. It is easy to check that (3.4) does not depend on the choice ofpresentations of and and so the multiplication is well-defined. Clearly,
and
$v$

$v$

$w=\Sigma_{s\in S_{\mathrm{e}}}\beta_{\delta}\delta_{s}$$v=\Sigma_{t\in S_{\mathrm{e}}}\alpha_{f}\delta_{t}$$v\cdot w=w\cdot v$$w$$||v \cdot w||_{X(\mathrm{I}\})}N\leq\sum_{t,s\in S_{\epsilon}}|\alpha_{t}|\cdot|\beta_{\epsilon}|\leq||v||_{X(\mathrm{I}))}N$$w$.

$||w||_{X(\mathrm{D}^{N})}$.

24

All other axioms from the definition of a Banach algebra are trivially hold. Noticeon hasalso that for every$z\in \mathrm{D}^{n}$$K_{z}(v):=\delta_{z}\cdot v$,$v\in X(\mathrm{D}^{N})$

.

(3.5)

Thus we can define a bounded linear map $K$ :, by the formula
Banach space of bounded linear operators on$X(\mathrm{D}^{N})$—$B(X(\mathrm{D}^{N}))$, to the$X(\mathrm{D}^{N})$$K(v):= \sum_{t\in S_{e}}\alpha_{t}K_{t}$

,

(3.6)

is a presentation of . (Note that the above formula does notdepend on the choice of the presentation.) Then we have
where

$v$

$v= \sum_{t\in S_{\mathrm{c}}}\alpha_{t}\delta_{t}$is an isometric embedding

Proposition 3.12 The map $K$ :and homomorphism of Banach algebras.

$X(\mathrm{D}^{N})arrow B(X(\mathrm{D}^{N}))$

with its image under $K$ we can naturally complete it to the
Identifying
Banach algebra with unit, just adding to this algebra the one-dimensional vector
generated by the identity map .
subspace of$X(\mathrm{D}^{N})$$I$$B(X(\mathrm{D}^{N}))$

generated
Remark 3.13 Consider the minimal closed subspace $X$ in$X$. Then from Proposition 3.12 it follows that is a Banach algebraby operators. In this way one obtains an inner description of
isomorphic toonly. Everyin terms ofdetermines a continuous linear
defined by the formula $f(x):=[x(f)](1),$ $x\in X$ , where
functional on)is the unit. (Observe that every $x(f)$ belongs to$B(H^{\infty}(\mathrm{D}^{N}))$$K_{z}^{*}$$X(\mathrm{D}^{N})$$X(\mathrm{D}^{N})$$f\in H^{\infty}(\mathrm{D}^{N})$$H^{\infty}(\mathrm{D}^{N})$$X\cong X(\mathrm{D}^{N})$$1\in \mathrm{T}^{N}$$H^{\infty}(\mathrm{D}^{N})\cap C(’\mathrm{F}^{N}).$is the set of functionals
Now the set of nonzero complex homomorphisms of, whereare the coordinate functionals. Next,is then identified with. The set of maximal ideals ofonis the bounded function
on
the Gelfand transform of every
defined by$X(\mathrm{D}^{N})$$\{z_{1}^{\alpha_{1}}\cdots z_{N^{N}}^{\alpha} :\alpha_{k}\in \mathbb{Z}_{+}, 1\leq k\leq N\}$$\mathbb{C}^{N}$$z_{1},$$\ldots,$$z_{N}$$X(\mathrm{D}^{N})$$(\mathbb{Z}_{+})^{N}$$v\in X(\mathrm{D}^{N})$$f_{v}(\alpha_{1}, \ldots, \alpha_{N}):=z_{1}^{\alpha_{1}}(v)\cdots z_{N^{N}}^{\alpha}(v)$The function$f_{v}$$f_{v}$,$(\alpha_{1}, \ldots, \alpha_{N})\in(\mathbb{Z}_{+})^{N}$$(\mathbb{Z}_{+})^{N}$.

also satisfies$\lim$$f_{v}(\alpha_{1}, \ldots, \alpha_{N})=0$$( \max\alpha_{i})arrow\infty$

The Gelfand transform
phism of Banach algebras. However, the image of
Also, we have
$F:X(\mathrm{D}^{N})arrow \mathrm{c}_{0}((\mathbb{Z}_{+})^{N}),$$\lim_{narrow\infty}$

Il.

$v\vdasharrow f_{v}$, is an injective homomorunder is not closed!

$X(\mathrm{D}^{N})$$v^{n}||^{1/n}= \sup_{\alpha}|f_{v}(\alpha)|$

$F$

.

:
is holomorphic, and thethe map
For everyis. Further, let us consider the linear
map
is an embedding if and only if
thebyformula:ondefinedbounded functional$v\in X(\mathrm{D}^{N})$$F\circ\psi_{v}$$\overline{\mathrm{D}}^{N}arrow c_{0}((\mathbb{Z}_{+})^{N})$$F\circ\psi_{v}$$\psi_{v}$$l$$l_{1}((\mathbb{Z}_{+})^{N})$$l( \{v_{\alpha}\}_{\alpha\in(\mathrm{z}_{+})^{N}}):=\sum_{\alpha}v_{\alpha}$.

25. It is a
In the next result we use the definition of the Marcinkiewicz space(with respect to the Lebesgue
quasi-Banach space of measurable functions on
measure) satisfying$M^{1/N}(\mathrm{T}^{N})$$\mathrm{T}^{N}$$f$$mes \{z\in \mathrm{T}^{N} :The optimal for all
have$\lambda>0$|f(z)|\geq\lambda\}\leq\frac{C}{\lambda^{1/N}}$constant,$\lambda>0$

in such inequalities is the norm of . Then we

$C$

$f$

Proposition 3.14 For eachis holomorphic and is extended tolinear map$z\in \mathrm{D}^{N}f(F\mathrm{o}\psi_{v})(z)\in l_{1}((\mathbb{Z}_{+})^{N})$$\mathrm{D}^{N}arrow \mathbb{C}$$M:X(\mathrm{D}^{N})arrow M^{1/N}(\mathrm{T}^{N}),$$\mathrm{T}^{N}$, the function

:
. The

$l\mathrm{o}F\mathrm{o}\psi_{v}$as a function from, is continuous and injective.
$M^{1/N}(’\mathrm{F}^{N})$$v\mapsto l\mathrm{o}F\mathrm{o}\psi_{v}|_{\mathrm{F}^{N}}’$

Rom this result we obtain another description of. Observe that the Cauchy projector
plex spaceextended toand then its image belongs to$X(\mathrm{D}^{N})$$L^{1}(\mathrm{T}^{N})$$C$:$C(L^{1}(\mathrm{T}^{N}))$. Consider the com-$L^{2}(\mathrm{T}^{N})arrow H^{2}(\mathrm{D}^{N})$$M^{1/N}(\mathrm{T}^{N})$$L^{1}(\mathrm{T}^{N})$

Proposition 3.15 The image
rally isometric to
.

.

.

is

equipped with the quotient norm is natu-$X(\mathrm{D}^{N})$(Similar to the above proposition results are valid also for other spaces $X(M)$ witha bounded domain.)
Finally, we recall that the Hadamard product of two formal power series$M\subset \mathbb{C}^{N}$,$f_{i}(z)= \sum_{\alpha\in(\mathrm{z}_{+})^{N}}a:,\alpha z^{\alpha}$on$\mathbb{C}^{N}$

where$z^{\alpha}$$:=z^{\alpha_{1}}\cdots z^{\alpha_{N}}$

for

$:=$

$\alpha$$i=1,2$ ,$(\alpha_{1}, \ldots , \alpha_{N})$is the series$\mathcal{H}(f_{1}, f_{2})(z):=\sum_{\alpha\in(\mathrm{z}_{+})^{N}}(a_{1,\alpha}a_{2,\alpha})z^{\alpha}$.

For formal Fourier series on
the Hadamard product coincides with the convolution of.
Proposition 3.16 The operator
satisfies$\mathrm{T}^{N}$$\mathrm{s}e\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{s}*$$M:X(\mathrm{D}^{N})arrow M^{1/N}(\mathrm{T}^{N})$$M(v_{1}\cdot v_{2})=M(v_{1})*M(v_{2})$.

In particular, forwhere$||\cdot||$$f_{1},$$f_{2}\in C(L^{1}(\mathrm{T}^{N}))$

we have,$v_{1},$$v_{2}\in X(\mathrm{D}^{N})$.

$f_{1}*f_{2}\in C(L^{1}(\mathrm{T}^{N}))$and$||f_{1}*f_{2}||_{M^{1/N}(\mathrm{T}^{N})}\leq||f_{1}*f_{2}||\leq||f_{1}||\cdot||f_{2}||$is the quotient norm induced from
$M$

: (X

$(\mathrm{D}^{N}),$$\cdot,$$|\cdot|_{X(\mathrm{D}^{N})}$)$L^{1}(\mathrm{T}^{N})$by C. Therefore$arrow(C(L^{1}(\mathrm{T}^{N})), *, ||\cdot||)$is an isomorphism of Banach algebras.
Moreover, for
we have$f\in C(L^{1}(\mathrm{T}^{N}))$$\lim_{narrow\infty}||f*\cdots*f||^{1/n}=\sup_{\alpha\sim,n\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{s}}|a_{\alpha}|$are Fourier coefficients of .
Similar results are valid for functions from different Hardy spaces.

where$a_{\alpha}$

$f$

26

References[Brl] A. Brudnyi, Projections in the space
and the Corona Theorem for coverings of bordered Riemann surfaces, Arkiv f\"or Matematik, 42 (2004), no. 1,$H^{\infty}$

31-59.

[Br2] A. Brudnyi, Grauert and Lax-Halmos type theorems and extension of matrices, J. Funct. Anal., 206 (2004), 87-108.
with entries in$H^{\infty}$[Br3] A. Brudnyi, Matrix-valued corona theorem on coverings of Riemann surfacesof finite type, Preprint 2006, University of Calgary.
[C1] L. Carleson, Interpolation by bounded analytic functions and the corona problem, Ann. of Math. 76 (1962), 547-559.

[C2] L. Carleson, On
in multiply connected domains, Conference on harmonic
analysis in honor of Antoni Zygmund, Vol. II, ed. W. Beckner, et al, Wadsworth,
1983, pp. 349-372.
$H^{\infty}$[F] F. Forelli, Bounded holomorphic functions and projections. Illinois J. Math. 10(1966), 367-380.
[G] J. Garnett, Bounded analytic functions. Academic Press, New York, 1980.
[GJ] J. B. Garnett and P. W. Jones, The corona theorem for Denjoy domains, Acta
Math. 155 (1985), 27-40.

[Gr] H. Grauert, Analytische Faserungen \"uber Holomorph Vollst\"andigen R\"aumen.
Math. Ann. 135 (1958), 263-278.
[JM] P. Jones and D. Marshall, Critical points of Green’s functions, harmonic measure and the corona theorem. Ark. Mat. 23 no.2 (1985), 281-314.
[L] V. Lin, Holomorphic fibering and multivalued functions of elements of a Banach
algebra. Funct. Anal. and its Appl. English translation, 7(2) (1973), 122-128.

[Ni] N. Nikolski, Treatise on the shift operator, Springer-Verlag, Berlin-New York,1986.
[T] V. Tolokonnikov, Extension problem to an invertible matrix. Proc. Amer. Math.
Soc. 117 (1993), no.4, 1023-1030.