U NIVERSI DAD AUTO N OM A METRO PO LITANA Casa abierta ai tiempo UNIDAD IZTAPALAPA División de Ciencias Básicas e Ingeniería

DEPARTAMENTO DE MATEMATICAS Area de Aridisis

MULTIVARIABLE SPECTRAL THEORY OF ALGEBRAS OF ANALYTIC FUNCTIONS

Autores:

I

Generao Almendra (U.A. Chapingo) Antoni Wawrzyñczyk (U.A.M.)

L 04.0402.I.01.002.94

1

Multivariable spectral theory of algebras of analytic functions

Genaro Almendra * .. Universidad Autónoma Chapingo

Area de Matemáticas and

Antoni Wawrzyríczykt Universidad Autónoma Metropolitana-Iz t apalapa

Departamento de Matemáticas

March 1, 1994 f

1

Abstract

The concept of the extended cospectrum C(J) of an ideal J C dp(L?) is introduced. One proves that the extended cospectrum is nonvoid for any proper ideal J. We discuss the joint spectrum of a

its relation with c(J). The spectral mapping theorem is proved for this type of spectrum. Basic properties of the generalized resolvent are established.

system of elements f j E dp(sl), j = 1,. ... k relative to J establishing

Introduction The present paper is devoted to the study of cospectra of ideals J of algebras d(S1) of holornorphic functions with restricted growth defined in an open set

*Partially supported by CONACyT +Partially supported by SNI

1

. . . . . - - ................ . . . . . . . . . . . ... .. __- . I

R c C'. We consider also the joint spectra of Ic-tuples of elements of the quotient algebra d(R)/J.

The principal innovation"is the addition to the classical cospectrum of "points at infinity". The obtained extended cospectrum is briefly speaking the set of common zeroes of the functions f E J extended continuously to the Stone-Cech compactification PR of R. The extended cospectrum is then a subset of PO although we consider later also the cospectra in other compactifications of R.

The appearence of ,OR in the spectral analysis of algebras of continuous functions on R is not surprising. In the case of the algebra C(R) of all continuous functions on R the theorem of Gelfand-Kolmogoroff identifies the space PR with the set of the maximal ideals of the algebra. To the point z E Ps2 there correspond under this identification the ideal

Mz = {f E C(R)l z E z(f)}. In the above formula Z ( f ) denotes the set of zeroes of the function f and the closure is taken in the compact space PO.

In case of the algebras of functions with restricted growth the relation between the p c h s of and maximal ideals must be modified. Briefly speaking we treat the cospectrum of an ideal of the form fd(0) as the obstacle to the invertibility of f. Even if the function l/f makes sense in some infinite region of R the invertibility depends upon the behaviour of this function at infinity which is restricted by the growth condictions defining the algebra d(R). A point z E is cospectral ( z E c( J ) ) if for every neighbourhood O of z there exists an element f E J such that l/f does not behave in O as elements of d ( R ) do. In the same way we define the cospectrum <(J7 C) of J in an arbitrary compactification C of R.

First of all we succeed in proving that nontrivial ideal J have nonempty cospectrum for an arbitrary compactification C.

It results that ((J) = <(J,PR) determines ( ' (J ,C) for an arbitrary com- pactification C. If Pc: ,OR + C is the natural projection which leaves invari- ant the points of R then

C ( J 7 C ) = PCC(J>.

If we associate to a point z E pin the ideal

2

we can obtain all maximal ideals of d(S2) as M, for appropiate z E @O. The continuous multiplicative furrctionals of the algebra A( S2) correspond

however to the points of 0 and are of the form f --f f ( z ) with z E S2. Our interest in the algebras of the form A(R)/J is motivated by the role

played by this type of algebras in the spectral analysis of translation invariant function spaces 0n-R". In p+rticular if V c C'(wn) is a linear translation

.invariant ciosed subspace the annihilator

V I = {T E C"(w")'l T ( f ) = o, f E V }

Ap(W = {f E A(c")I Ifk)l 5 Aexp(B(log(1 + 1 1 ~ 1 1 ) + IIImzll))),

is isomorphic by means of the Fourier transform to an ideal J of the algebra

while the dual space V' is isomorphic to the quotient algebra Ap(C)/J. This being the case an exponential function e,:R" 3 x + exp((z1x))

belongs to V if and only if z E ('(J). Using the traditional terminology the spectral analysis holds in V if C(J) n UY is nonvoid.

On the other hand it can be seen that the set c(J) n UT is equal to the joint spectrum of the n-tuple 2 = ([zi], . . . , [z,,]) E d(P)/J)". In section 3 we introduce the notion of the extended joint spectrum and we prove that the latterjis e q h just to ('(J) in case of the n-tuple 2. Finally we see that both the study of the generalized cospectrum <(J) as well as of the extended joint spectrum can be treated as a subsequent step in the development of the spectral analysis and synthesis.

The mentioned extended joint spectrum of a Ic-tuple .F = ( [fi], . . . , [ f k ] ) E (A(0)IJ)" is defined as a closed subset of an arbitrary compactification Ií of <Ck and i s denoted by g(.F, J, K). Our definition assures that in @'\\a(F, J, I í ) there exists a generalized resolvent that i s a IC -tuple of functions R3(z,p) such that

I

k

j=l

for p E Ck \ u(F, J, K). Moreover, for every X E Ií \ a(F, J, I() there exists a neighbourhood O of X such that the set { & ( e , p)l p E O ri ( C k } is bounded in A(R).

It results that the joint spectrum in the Stone-Cech compactification BCk determines joint spectra in other compactifications:

43, J, I - ) = P K O ( 3 , J, p9).

3

',

..... * . ~ . .. . . . . . . . , . . . . ~ . . .

In what follows we write just c(F, J ) in place of c(3, J, @).

that The joint 2pectrum is never-empty for proper ideals J because we prove

-F(C(J)) c 4 3 7 J)

PKF(C(J-)) c c(3, J7 K ) .

'1.

and

In the formula above 3 denotes the unique continuous extension to pfl of the mapping fl 3 z + (fi,. . . , fk) E Be.

In the case when K is equal to the real or complex projective space P 2 k ( ~ ) or Pk(@) we have the equality

-.

PKF(C(J)) = c(3, J, K).

In both cases the spectrum e(F7 J, K ) have the spectral mapping property what means the following:

For every polynomial mapping P: Ck -f P which extends to a continuous application @: P k ( C ) -+ P"(C) we have

1 f @a(3, J, P") = c (P 0 3, J, PY).

The analogous formula for the real projective space is also valid. In the last section we prove that all multiplicative functionals on d,(fl)

are the evaluations of f E &(O) at a fixed z E R. The present paper presents the results of an investigation not finished yet.

The interest of the authors concentrates at this moment at those properties of ideals which can be expessed in terms of their cospectrum C(J). It can bee seen for example that the slowly decreasing n-tuple 3 (see [l] for the definition and fundamental properties) whose cospectrum consists uniquely of »points at infinity" generates d,(R). It explains the fact that proper ide- als generated by slowly decreasing tuples have always nontrivial the classical cospectrum. The slowly decreasing tuples were created as those which gen- erate closed ideals. This property is also related with the appearence of the points at infinity in ((JF).

It is easy to see that the ideal fdp(R) is closed if and only if f is a topological divisor of zero in dp(fl) exactly as in the case of commutative Banach algebras without divisors of zero. Probably "the points at infinity" form the Silov boundary of A,(R2) and the corresponding maximal ideals are

4

composed of topological divisors of zero. (See [13] for the analogy in the theory of commutative Banach algebras.)

It is worthy of mention that the development of the funtional calculus for the algebra d,(R)/J leads to the interpolation theorems similar to those obtained in [l].

2 Preliminaries In what follows R will denote an open subset of CF and p a function on R which is positive plurisubharmonic and satisfies the following conditions

1. log(l+ 11 z ll)/p(z) is bounded on R. 2. there exist A,B,C,D > O such that for all z E R the

condition 11 z - w I [ < exp(-Cp(z) - O) implies u] E R and P ( W ) 5 Ap(z) + B.

In case of R = C the above conditions are satisfied for p(z) = IzI', T > O as well as for the function p(z) = log(1 + 1 1 ~ 1 1 ) + llIm zll which was already introduced in tqe previous section. If R is a domain of holomorphy we obtain a function satisfying 1 and 2 putting p(z) = - log(d(z, aR)), where XI is the boundary of R and d denotes the Euclidean distance in Cn.

This type of pairs (R,p) was introduced by Hormander in [6] with the purpose to study the finitely generated ideals of the algebra of functions which are holomorphic in R and have its growth determinated by p.

We denote by C the space of all continuous functions on R and by d ( R ) the space of functions holomorphic in R. For T > O let Ci denote the set of continuous functions which satisfy

and Cp( 0) = Cp. We also introduce Al; = d(R)nCp(R) and d p ( R ) = Ur0 di. In the spaces

d p ( R ) and C,(fl) we define the inductive limits topologies of the normed spaces (di, 11 - Ilr) and (Ci, 11 . [Ir) respectively.

The spaces d p ( R ) and C,(R) are topological algebras which constitute the principal subject of our research.

As proved in [6] the conditions 1 and 2 assure that

5

i. The space of all polynomials belongs to &(O). ii. Iff E dp(n) then & f E d, (O) for any 1 5 j 5 n.

The following theorem of Hormander (and its extension given later) is the fundamental argument used to develope the notion of the cospectrum and joint spectrum in algebras &(O). *

Theorem 2.1 [6] Suppose that the fdnction p > O on O c C is plurisub- harmonic and satisfies the conditions 1 and 2. Then the ideal generated by the functions fi, . . . fk E dp(0) is the whole d,(O) if and only if there exist cl, c2 > O such that

k

j=i

The proof of the above theorem uses the celebrated results of Hormander about the solution of the equation ah = g in spaces of differential forms with measurable coefficients which behave at infinity as elements of C,(n).

In case of the algebra C,(O) the above theorem is trivially valid and its proof is just the first step in the Hormander's proof. Given the functions f1 , . . . , f k , E e,(.) which satisfy the condition(2) we can construct

- c.

Thanks to the condition (2) and the relation xtZl I fjl 5 IC'I2(Ct=, lfj12)'/2

we see that hj E C,(O). The direct calculation gives hj f, = 1, hence for arbitrary 4 E C,(n) we have 4 = xi=l f j ( h j 4 ) . The ideal generated by the functions {fj} is just C,(O).

We shall need however a strengthened version of Theorem 2.1 provid- ing several informations about the coefficients hj E d , ( R ) which permit to generate the function 1 from F. *

For given 3 = ( f ~ . . . fk) let us denote

In [12] the following result is proved:

6

~_.___ ._I_ ._ . . . ... .-.. - - - ..... ~~ . . . . -

Theorem 2.2 For every k E N and c 2 O there exist t , r 2 O and a poly- nomial Wk.3 with positive coeficients such that for every g E A; and every k-tuple 3 E Ap(0)k which satisfies the condition (2) and Ila3ll. 5 00 there exist hj E di obeying

k

x h j f j = 1 j=i

and

3 The cospectrum In what follows we present a number of definitions and results parallely for the algebras C,(R) and Ap(R). In order to simplify the exposition we understand by A anyone of this algebras. Let ( C , j ) be a compactification of R, that is to say C is a compact completely regular space and j: R -, C is a continuous injection with dense image.

Definition 3.E For a given ideal J E A and a compactification C of R we denote

p( J, C ) = { z E C13 f j E J, K > O and a neighbourhood O o f z in C

such that I f j (w) l 2 exp(-Kp(w)) for all w E O n R}. The points belonging to p( J, C ) are called regular points for J in C. The complement of p( J, C) is called the extended cospectrvm of J in C and is denoted by ((J, C ) . We denote by cosp J the classical cospectrum of J that is the set of common zeroes of the elements of J in R.

It is easily seen that cosp J = (( J, C) n R independently of the particular compactification of R. Theorem 3.1 The set <( J, C) is empty if and only if J = A.

I( > O and f ; E J such that Proof. Suppose that p(J,C) = C. For every z E C there exist O ( z ) ,

kz

Ifj”(4l 2 exp(-1{p(w)), E O(z) n R. j=l

7

By the compactness of C we can choose a finite subcovering {O( zi)}*=l,...,,

from the covering { O ( Z ) } , ~ C . Let {f?},. . . , {f?} and K1,. . . ,K, be the corresponding elements of J and cofresponding positive constants. Taking c = max( Kl, . . . , K,) we obtain

i = l j=i

By Theorem 2.1 (or its version for C(R)) the functions f?, . . . , f34f generate A, that is J = A. The converse is obvious.

O In contrast to (( J, C) the classical cospectrum can be empty as shows the

The family Comp R of all compactifications of R has its natural order example found by Gurevich [5].

defined as follows:

provided that there exists a continuous surjection P:C1 -+ C2 such that j , = P o jl. The maximal element of Comp R with respect to the order >- is the Stone-Cech compactification PO which is unique up to homeomorphism.

We denote: k( J ) = (( J, BR) and p( J ) = p( J, PO). The Stone-Cech compactification can be defined equivalently up to a

homeomorphism as the compact space containing R and such that any contin- uous map $: -+ K valued in a compact space K can be uniquely extended to a continuous map 6: PR + K . This property of PR permits us to describe the cospectrum of J as the zero set of a family of functions associated to J. Theorem 3.2 Let J be an ideal of A. Then

(C1,jl) + ( C 2 7 j 2 )

((J) = { z E PRI (fexp(cp))-(2) = Ofor all c > O}. Proof. If J is a proper ideal and z E [( J ) then by the very definition of the cospectrum for arbitrary E > O, c > O and every neighbourhood O of z there exists w E O such that

I f ( 4 exp(cp(w)> 5 e-

By the continuity of the extension we obtain the assertion. The converse is obvious. 0

The description of the cospectrum obtained above implies in particular the following:

. . ~r . ...,-........ . ~ . - . ..... . . .. - .. .

Theorem 3.3 Let I , J be proper ideals of A, and let ,? be the ideal generated by I and J. Then

((a = W) fl C( J > - Proof. Assume that z E ( ( I ) ri [( J). For every f E J there exist g E I , h E J and c p , $ E A, such that f = vg+$h. There exist also constants u , b, c, d > O for which 191 5 uexp(bp) and I $ [ 5 cexp(dp). For an arbitrary y > O we obtain

Ifexp(7p)I 5 491 exp((b + YIP) + clhl exp((d + Y)P)-

Both terms of the last sum are continuous functions which by Theorem 3.2 extend to functions vanishing at z. Applying the same theorem we obtain z E [(g). We have proved that

Since J C J and I C J the opposite relation is also valid. 0

In case of the algebra C,(R) the description of the cospectrum can be simplified because for f E J the function fexp(cp) belongs to J. We obtain:

( ( J ) = { z E POI f(z) = O, f E J}. 1

It follows immediately from this observation that the cospectrum of J C A,(R) is equal to the cospectrum of the ideal of C,(R) generated by J. In particular if we consider the cospectrum of an ideal JF generated by several elements fi,. . . , fk E d,(R) it does not matter in which of two algebras we generate the ideal. By this reason we have decided to simplify the notation of (( J) the cospectrum avoiding to anote the algebra in question.

It is important to observe that the cospectrum ((J) of J determines com- pletely the cospectra of J in other compactifications as shows the following

Theorem 3.4 Let C be an arbitrary compactification of C! and let P : BO --t C be the projection which leaves invariant the points of R. Then for every ideal J c A

PK(J>> = ( (J7 0- (3)

9

. .~ . . . . . .

Proof. We prove the relation P(<(J)) L ((J, C) by showing that P ( z ) E p( J , C ) implies z E p(J) : In fact, if O(P(z) ) , f j E J , c > O are such that

j=i

for w E O n R, then U = P-'(O(P(z))) is the neighbourhood of z such that the same inequality is valid for the points of U n a. It means that P-'(p(J,C)) c p(J), or equivalently P(c(J)) C <(J,C).

In order to prove the opposite inclusion, we assume that z E c(J, C) but P"(z) C p ( J ) . This will lead to a contradiction.

Suppose that for every IC E P-'(z) there exist f j E J and e, c > O such that (E:=, Ifj(y)I exp(cp(y)))- > E in some neighbourhood of IC. Since P-'(z) is compact we can find an N-tuple f 1 , . . . , f~ E J and E > O, c > O such that

for all IC E P-' (9).

of z in C there exists zu E U n 0 such that On thi: othe;' hand if z E [(J, C) we know that in each neighbourhood U

N

i=l

By the compactness of PR the net (zu) has an accumulation point z which must belong to P-'(z) by the definition of the net. It implies that (ELl Ifil exp(cp))"(s) 5 e/2. This is the contradiction.

U The description of the maximal ideals plays the fundamental role in the

spectral theory of topological algebras. If J is a maximal ideal of A (not nec- cessarily closed) and z E <(J) then by the very definition of the cospectrum J c J, = {f E APl(fexp(Cp))-(z) = O for all C > O}.

The set J, is a proper ideal of A hence by the maximality of J both ideals coincide. It follows that all maximal ideals of A are of the form J, for z E PO. The question arise if ((J,) = { z } and if the ideal J, is maximal for each point z E PO. It is true for the algebra C,(O) because continuous

10

. - . . . . .... ,...__ . . . . . . , . .__. _ _ _ _ _._.. . . - . . ,. . . . - .

bounded functions separate points of PR. This being the case we obtain the identification between the space PSZ and the space M(C,) of all maximal ideals of C,(R). In generd we can only state that there exists an injection of the space M(A) into the space of equivalence classes pR/- where for z , w E PR we define Z-w if f(z) = f ( w ) for all f E A. It remains to establish if this mapping is onto. Let us note that PRl- provided with the quotient topology is also a com$ctification of R and the elements of A have unique continuous extensions to PI-.

*

4 The quotient algebra d/J Let us consider an ideal J c A and the quotient algebra A/ J . The invertibil- ity of an element [ f ] E d/J can be expressed in terms of the corresponding cospectra.

Theorem 4.1 Let f E A. The following conditions are equivalent:

a) The class [f] is invertible in dIJ.

q) 3c 5 O such that ( f exp(cp))- does not vanish on [(J). Proof. The class [ f ] is invertible if there exist g E A and h E J such that f g - 1 = h. Equivalently, the ideal generated by f and J is equal to A. Its cospectrum is empty and by Theorem 2.5 it is equal to c ( J j ) f~ <(J). This proves u) + b).

Now, suppose that c) is not valid, that is Vc > O the function (f exp(cp))- vanishes somewhere in (‘(J). Denote by A, the (compact) set of zeroes in [ ( J ) of the function (fexp(cp))-. Obviously c < d implies Ad c A,. By the compactness of [(J) the set o,,, A, C (‘(Jf) n (‘(J) is nonvoid. We proved that a) + c).

Finally assume that the condition c ) is satisfied and consider the ideal J generated by f and J, whose cospectrum is [(Jf) n [(J) according to Theorem 2.5. If J is nontrivial and z E [ ( J ) then in particular for every c > O the extension of the function fexp(cp) vanishes at Z . This contradicts c) and the invertibility of [ f ] is proved.

b) (‘(Jf) n( ‘ (4 = 0-

o Let us compare the situation with the classical model of the Banach

algebra of C(X) of continuous functions on a compact set X. If J is a closed

11

ideal of C(X) and C( J ) is the set of common zeroes of the elements of J then the invertibility of [ f ] in C ( X ) / J is equivalent to the condition that f does not vanish on ((J) or equivalently that <(Jf) n ((J) = 0. In this case the spectrum of [ f ] in the quotient algebra coincides with the set of values of f on <(J). The set ( ( J ) plays also the role of the spectrum of the quotient algebra if the latter concept is defined as the set of multiplicative functionals on C(X).

Now we pass to define the extended spectrum of elements of the algebra A. To avoid repetitions of arguments we define at once the extended joint spectrum of a k-tuple of elements in an arbitrary compactification A' of C k .

Definition 4.1 Denote by 3 a k-tuple ( f i , . . . , f k ) of elements of A. Let X E I(. We say that X is regular for 3 relative to J if there exist c > O , d > O and a neighbourhood O(X) of X in Ií such that for all p E O(X) n Ck and

k f C ( J >

The set of regular points for F in I( is denoted by p ( 3 , J, I(). It is open in I(. Its complement is called the joint spectrum of 3 in K with respect to J and is denoted a(3, J, K ) . We denote by ~(3, J ) (resp. e ( 3 , J ) ) the spec- trum (resp. the set of regular points) of 3 in the Stone-Cech compactification

Let us observe that the sets e(F, J,K) and ~(3, J,K) depend in fact only on classes [F] = ([fi], . . . , [ f k ] ) hence we deal with an extension of the concept of the joint spectrum of a k-tuple of elements of A/J. First of all we shall prove that for all finite points p7s in some neighbourhood of each regular element X E íí there exists a generalized resolvent R(p) and the set of all R ( p ) g , E ~ is bounded.

,

Bck.

Theorem 4.2 Let X E e ( 3 , J,K) and let O C Ií be a neighbourhood of X in which the inequality (4) is valid for several c , d > O . Denote U = O ri C k . There exist functions Rj( . ,p) E A, p E U such that

k

j=i

12

',

~ . .. . _ _ .. . . .. . . . . - . . . . - . .

for all /-I E U. There exist r > O, b > O such that l lR,(.,p)/lT 5 b for all p E U.

Proof. If z 4 C( J ) then there exists gz E J and c, > O such that

I s z ( w ) I > exP(-c,P(4)

for all w in some neighbourhood of z in R. By the supposition that X'is regular for 3 'we obtain that

k

1 l f j ( 4 - PJIeXP(c(P(w)) 2 d j=l

for appropriate c > 0,d > O and for all w E R in some neighbourhood of

we can choose a finite number of functions C( J).

By the compactness of 91, ...,gn E J such that for some c , d > O and for all ( w , p ) E R x U

k m

j = i i=l

By Theorem 2.1 (or its version for C(R)) the functions

{ f j - p j , g i } i < j < k , i < i < m

generate the space A. In particular there exist functions Rj E A such that

k k+m

j=i j=k+i

Since the second sum is an element of J we obtain the formula ( 5 ) . The second part of the assertion follows by Theorem 2.2 in case of the algebra d,(O) and directly by the definition of the resolvents defined in section 2 when the algebra C,(R) is considered.

o By the usual application of a partition of unity we can construct easily a

global resolvent of the class C"(R). Nevertheless the nonuniqueness of the resolvent makes difficult the study of its global properties.

The equation (7) permits us to formulate the condition of regularity of a point in a slightly stronger form.

13

Corollary 4.1 Let Ií be a compactification of C k . A point X E K is regular for 3 relative to J if and only zf there exist a neighbowhood O(X), d > O , c > O and a neighbourhood V of [(J) such that (4) is valid for all z E V and for all p E O(X) n ck. Proof. Assume that X satisfies all conditions determined in Definition 4.1. Let c > O, u > O be such that the functions R,(w, p) in (7) satisfy IR,(-, p) I 5 uexp(cp) for all finite p E #(A). Next define

1 k+m

v = (2 E pfll('exp(cp) lgJ-k l )"(z ) < 5). j=k+l

The set Y is a neighbourhood of [(J). Thanks to (7) we obtain for every finite p E O(X) and for z E V :

k

Choosing d < & we obtain the desired result. o Exactly as in the case of the cospectrum the knowledge of the spectrum

~ ( 3 , J ) permits us to obtain c(F, J, Ii') for an arbitrary compactification Ií by simple projection. Let us denote by PK: PCk -+ Ii' the natural projection which leaves invariant the points of C k .

Proposition 4.1 PK(c(F, J ) ) = c(F, J, K) .

Proof. Suppose that X E Ií is regular for 3 relative to J. Let #(A) be a neighbourhood of X such that (4) is satisfied. Then Pcf(O(X)) is the neighbourhood of an Pc'(X) for which (4) is also satisfied. It means that all elements of P;'(X) are regular and consequently P~(a(3, J)) c a(3, J, IC).

Now assume that X E ~ ( 3 , J, I<). For all d, c > O and for each neighbour- hood O of X there exist po E O n Ck and zo E a(J) such that

\

14

',

...... ..... .~ . .......... - .-- . . . . . . . . ..... ~ .

By the compactness of pCK the generalized sequence {p'} has an accumu- lation point, say 1-1 E ,@Ek. It follows by the construction that Ph'p = A.

The point p is singular because at least for a subsequence pu which converges to 1-1 we have

k

Any k-tuple 3 = (fly. . . , f j ) of elements of A can be treated as a contin- uous mapping

0 3 z 3 ( f l ( z ) , . , f k ( z ) ) E ck c pc'. There exists a unique continuous extension of this map on the domain p0 which is denoted in the sequel by 3. The following result determines the relation between F([(J)) and a(3, J ) .

Proof. Let X = F ( w ) where w E [(J). Suppose that X is regular for 3 with respect to J that is

for d i z E [(J), p E O(X) n ck. By Corollary 4.1 the above inequality remains valid for p E O(X) and

z E Y where Y is a neighbourhood of [(J). Let us choose an arbitrary finite point z from Y n f-'(O(X)). Take po = F(z ) . Obviously po E O(X) but the left hand side of (9) vanishes at t . This is a contradiction.

O By applying to both sides of (8) the projection PK for an arbitrary com-

pactification I( of Ck we obtain immediately

Corollary 4.2 PKF(C(J)) C ~(3, J, IC).

15

5 Towards the spectral mapping theorem

The question if the'inclusion (8) is in fact an equality remains open. In the general case we can only prove that a(3, J) \ F(((J)) C Pa? \ Ck that is to say inside Ck both sets coincide.

Theorem 5.1- a(F, J) fl ck = F ( ( ( J ) ) .

Proof. In virtue of Theorem 4.3 it remains to prove that a(F,J) fl Ck C F(C(J ) ) . If X E ~(3, J) r l Ck then for arbitrary c > O and natural n we can find p(") E B(X, +) and 2, E ((J) such that

k

The factor exp(cp) is no less than 1 in the whole compactified domain which implies

k

, j=l

The sequences {?(zn)} and { P ( ~ ) } are equivalent hence both of them tend to A.

Since the cospectrum [(J) is compact the sequence (z,,) has an accumu- lation point in some z E ((J). It means that y ( z ) = A. O

The above result means that if there exists X E o(F, J) \ F(í(J)) then for the finite elements of some neighbourhood of X the generalized resolvents do exist but they form an unbounded subset in A.

In the very special case of R = C" and 3 = i d p we have equality in (8). Let us denote by 2 the n tuple for which the i-th function is just f ; ( z ) =

2;. The corresponding mapping 2 i s the identity in , L I P .

Theorem 5.2 Let iR = C'. Then

a (2 ,J ) = a(J). (11)

c

Proof. The relation c(J) c o(2 ,J ) follows by Theorem 4.3. It remains to prove that o(2 ,J ) C c(J). Suppose that X E a (2 ,J ) . According to

16

Corollary 4.1 for arbitrary c, E > O and for every neighbourhood O of X and V of a(J) there exists z E V and p E O í l Cn such that

- n

J=1

Our aim is to prove that for every f E J and d > O, the extension (f exp(dp))- vanishes at A. In virtue of Corollary 2.5 in [l] there exist functions Q3 E A(@") such that

n

f(z) - f ( P I = QJ ( z , 11)(zJ - ~ 3 )

3=1

and for some constants C, D depending only upon f the inequality

IQ3(Z,P)l 5 cexP(D(P(4 + P(P>)

is valid on Czn. Recall that the function p has the property that for appro- priate A , B > O the relation IIz - ,u11 < 1 implies that p(p) < Ap(z) + B. Choose the neighbourhood V in such a way that I f exp(d(Ap + B))I < E in V . We obtain 5

n

If(.> - f ( ~ ) l e x ~ ( d p ( ~ ) ) 5 ~xP(&(P))

5 c exp((D + d)P(P) + DP(Z))

- < c exp((D + d)B) exp(((D + 4 A + D)P(Z))

I Q ~ ( ~ , P ) I I ~ ~ - ~ 3 1

p 1 n

I 3 - PJ I J = 1

n

123 - PJ I. 3=1

Since the constants A , B, C, D depend only on the functions f and p we can suppose that the constant c was chosen from the beginning as ( (D+d)A+D) and in place of 6 in (12) the value E / C exp( (D + d)B) has been used. In this way we obtain

Taking into accout that If(z)[exp(dp(p)) < If(z)(exp(Ap(z) + B) 5 E we obtain If(p)l 5 2~ for several p E O. Since the neighbourhood O is arbitrary we conclude: (fexp(dp))-(A) = O.

If(.> - f (P)lexP(~P(P)) 5 E -

17

',

O Now applying Theorem 3.4 and Proposition 4.1 we obtain

Corollary 5.1 For an arbitrary compactification K of C"

42, J, K ) = ((J, K). 1 4

Y

Coming back to the question of the equality in place of the inequality in (10) let us observe that for the compactification of Alexandroff íia of Ck we have in fact

P ~ - , ~ ( c ( J ) ) = g ( ~ , J, K a > - This equation asserts simply that Theorem 5.1 is valid and both sides can be unbounded only at the same time.

Let us observe also the following

Proposition 5.1 Assume that for certain compactification IC we have the equalit y

PKF(((J)) = o(F, J, I - ) .

PCF(C(J)) = a(& J, C ) .

and that IC + C. Then

Proof. It suffices to apply the projection PKC : K -+ C to both sides of the first equation.

O - We can also obtain the equality in (10) by imposing several topological

condition on the compactification Ii of Ck.

Theorem 5.3 Let IC be a compactification of Ck such that for all A, Y E K there exist neighbourhoods O of X and U of Y such that

18 I

Proof. In virtue of Corollary 4.1 it remains to prove that g ( F , J , K ) C PKF(C(J)). We know that €or finite number of points the inclusion is valid, hence suppose that X E a(F, J,K) \ Ck and that X # &F(C(J)).

According to our supposition for every w E &F(C(J)) there exist neigh- bourhoods O, of X and U, óf w as well as a constant E , such that Ilp-wll > e, €or all p E O, nCk and b E U, h C k . Since the set A = P K ~ ( (( J ) ) is compact we can.choose a finite subcovering from the covering {U,},,A obtaining in this way neighbourhoods

U = u U,, and O = n O,, isism isism

such that IIp - u11 > E = minls;sm{c,,} for all finite p E O and w E U.

( P K ~ ) - ’ ( U ) such that However by the singularity of X there exists p’ E O n Ck and z E

k (E Ifj - P’I>-(4 < E . j=l

By continuity t i e same is valid for several z’ E (PKF)-’(U) n Ck. This is a contradiction, hence X E PKF( c( J)).

o In particular it is easy to see that the projective spaces P”“(), P k ( C )

which are compactifications of Ck satisfy the assumptions of the above theo- rem. We obtain

Corollary 5.2 For an arbitrary 3 E dk and for Ií = P”(W) or Ii‘ = Pk(@) we have

PKF(-(c(J)) = g(F7 J, K).

Let us now observe that the spectrum c (F , J) has other properties which are characteristic for various types of joint spectra considered in the theoryof commutative and noncommutative Banach algebras.

The space /3Ck can be considered in a natural way as a closed subsetof the Ic-th Cartesian product PCx . . . xP@. As the natural injection we consider the continuous extension of the mapping: Ck c-, l-Jf=l ,f?C. Using this convention we can assert the following:

19

k

Proposition 5.2 k

1= 1

Proof. We assert only that if a point X E ,Bek projected in n<=, G( { f i } , J ) on the rn-th variable gives us Am which is regular for f m relative to J , then X is regular for 3. This is obvious by the definition of ~ ( 3 , J).

O Let P(m) = (Pi , . . ,. , P,) be a tuple of polynomials of k complex vari-

ables. Denote by ?(,) the continuous extension to ,8Ck of the mapping (21 , - 7 z k ) - 7 Pm(z)) E ,8r- Theorem 5.4 For every k-tuple 3 and for every P(m)

.p(m)(a(F, J)) c a#") o 3, J).

Proof. For simplicity we write just P in place of P(m). Given finite points z E 0 p E let us write

f

,Pi(F).- pi(Pi) = &io(F)(fi - Pi)o1 * ( f k - P k ) Q k , I+l

. where a = ( a l , . . . , ayk) is a multiindex, Qia are polynomials of k variables and I' is the maximum of the orders of the polynomials Pi. Let us choose A, B > O in such a way that

[Qio 0 31 I Aexp(Bp)

for all values of i ,a. Assume that X E ,BCk belongs to the spectrum of 3 relative to J. Given c, E we can find in each neighbourhood of Xand in each neighbourhood of <( J ) finite points p and z such that

k

We obtain

20

m r

3=1

This proves that @(A) E g(P o 3, J).

Using the terminology of the spectral theory of topological algebras the above result is the one-way spectral mapping theorem for ~ ( 3 ~ J).

If we consider the projective space as the compactification of Ck we can obtain a version of the complete spectral mapping theorem.

Assume that 9: Pk(C) + P"(C) is a continuous application which maps Pk(C) \ Ck into Pm(C) \ C" and such that its restriction to Ck is of the form P = (pi,. . . ,p,), where pi are polynomials of k complex variables. As a unique continuous extension of a mapping of polynomial type P satisfies the relation

7j o Ppqq = I+"(@) 0 P. Theorem 5.5 %et 3 E Ak. Then

+(U(3 , J,P"C))) = g(P 0 F,J,P"(@)) .

Proof. We calculate applying Corollary 5.2 and (13):

+(,(F,J, P"(@)>) = +(PPk(,)F(c(J)))

= Pp-(q(P O F(C(J))) = U(" O 37 J, Pm(C)).

O If we use the real projective space P"(R) as the compactification of Ci we

obtain in the same way the formula

P(u(F, J, P2k(W))) = a(? o F, J, P"(R)), (14)

valid for every

which is of complex polynomial type on Ck (canonically imbedded in P"(R)) and which sends the points at infinity into points at infinity.

P: P"(W) -f P"(R)

21

',

6 Muit iplicat ive funct ionals on 'Ap (a) The functional of the evaluation at a given point z E R

X, : A 3 f + f ( z ) E C

is a continuous multiplicative functional on d p ( R ) . It is natural to ask if the functionals of the formy, for z E R are the unique continuous multiplicative functionals on d(Q). The answer is positive and in [4] we can find this result. It is deduced however from the same fact which is the subject of our Theorem 2.3 and it seems that its proof in [4] is not complete. In this section we prove a stromger result and the proof is simpler. In particular it does not make use of the functional calculus of Waelbroeck.

It results that all multiplicative functionals on d p ( R ) are given by the evaluation at a fixed point z E R. In particular it means that all multiplicative functionals are continuous and the maximal ideals M, for t # R are of codimention > 1.

Let us begin with the following

Proposition 6,l There exists t 2 O and for each X E R there ezist

r ~ ( X ) r - - . , r n ( X ) E Al;

such that the set { T ~ ( X ) } ~ ~ ~ ~ ~ , X E R is bounded in d; and

n

- X,)r,(X + T O P ) exP(-P(X)) = 1, (15) ,=1

for all z,X E R. Proof. Let us consider the n + 1-tuple of functions

z1 -Xl,...,zn-Xn,Ke~p(-p(X))

for I' > O. Recall that according to the definition of the function p there exist positive constants A, B, C, D such that E;==, Izj - Xjl exp(Ap(z)) 5 exp( -B) implies p(X) 5 Cp(z) + D. Let us take I< > max(1, A , C,exp(D)}. Consider the set

T = { z E 0 1 C Izj - ~ j l exp(I(p(z))l I exp(-B)). n

j=l

22

Inside T we have

The first term is greater or equal to 1 for z E T. arbitrary z E R we have

(1 ~ z j -

It 'means that for an

n

l(exp(-p(X))exp(l(p(z)) 2 min{i,exp(-B)}. j=l

By Theorem 2.3 there exist constants t , R 2 O and a family of n + 1-tuples .;(A), . . .,.:(A), X E R of elements of A; such that Ilri(X)llt < R and

n C<zj - ~ j ) r ; ( ~ ) + ~;(x)(K exp(-p(X)) = 1. :j=i

If we define rj = r(i for j 2 1 and ro = K r ; the proof follows. o

Corollary 6 .1 There exist a family { $ J a } a E ~ which is bounded in some di and such that

for ail z E R.

Proof. It suffices to bserve that by (11) we have ro(X)(X) = we define ?,be = ro(a) for Q E R we are done.

o

Corollary 6.2 For every ,u E PR \ R there exists f E d,(R) such that

lim If(.)[ = OO. 2-u

23

',

Proof. If all functions rO(a) for Q E fl have finite limit at p we pass to the inductive construction of the function f. Let crn E R be a sequence converging to p. Define f l = 1.

If the functions f1, . . . , fiv have been already defined and the function

1 N

S N ( 4 = ,f*(.) I

t = l

has at the point p finite limit equal to SN we choose zmN+l such that S N ( Z m N + , ) < 2.5” and expp(zmN+,)) > 2 N + 3 S ~ . We define f ~ + i = ro(zjv+~). In this way we assure that ISN+1(zmN+l)1/2N+1 > 2lS~l. On the other hand he series

O O . 1

f = c , f i i= 1

is convergent in dp(Q) since the sequence {ro(ZN)} is bounded. It follows by the construction that I f (zmN)I > 2N-1 hence If(,u)l = m.

o

Theorem 6.1 :For every nonzero multiplicative functional x on A,(R) there exists z E R such that x = xz.

Proof. Let J = { f E d,(Q)lx(f) = O}. Since f - x(f) E J for an arbitrary f f dp(i2) the quotient dp(R)/J is isomorphic to C hence the closed ideal J is maximal. By Theorems 3.1 and 3.2 there exists p f PR such that for all f E J and c > O one has limz-p f( z ) exp cp(z)) = O.

In particular we have for an arbitrary f:

It means that each function in dp(Q) havs in p finite limit equal to x(f). By Corollary 6.2 it is possible only if p E R.

O

Corollary 6.3 All multiplicative functionals on dp( 0) are continuous. All maximal ideals of &(O) of codimension 1 are of the form M, where z E 0.

24

References

[l] C.A.Berenstein and B.A.Taylor, Interpolation problems in CY with ap- plications to harmonic analysis, J. Anal. Math., 38, (1980), 188-254.

[2 ] L.Carleson, Interpolation by bounded analytic functions and the corona problem, Ann. of Math. 2, 76, (1962), 547-559.

[3] L-Ehrenpreis, Fourier Analysis in Several Complex variables, Wiley- Interscience, New York, 1970.

[4] J-P.Ferrier, Spectral Theory and Complex Analysis, North-Holland, Mathematics Studies 4, 1973.

[5] D.I.Gurevich, Countresamples to the problem of LSchwartz, Funkt- sional. Anal. i Prilozhen.,2, 9, (1975), 29-35.

[6] L.Hormander, Generators of some rings of analytic functions, Ann. of Math., 2, 76, (1962), 943-949.

[7] L.Hormander, An Introduction to Complex Analysis in Several Vari- ables; Springer Verlag, 1990.

[8] R.C. Walker, The Stone-Cech Compactification, Springer Verlag, 1976.

[9] L.Waelbroeck, Lectures in spectral theory, Dept. of Math., Yale Univer- sity, 1963.

[lo] A.Wawrzyríczyk, How to make nontrivial the spectrum of a translation invariant space o f smooth functions, Informe de Investigación, UAhI-I, 1991.

[ll] A.Wawrzyíiczyk, El espectro extendido y el análisis espectral de espacios invariantes, Aport. Mat., Comunicaciones 11, (1992), 41-55.

[12] A.Wawrzyríczyk, A note on generating ideals of holomorphic functions, Informe de Investigación, UAM-I, 1993, submitted f o r publication.

[13] W.Zelazko, Banach Algebras, Elsevier Publ: Comp. and PWN, 1973.

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