On a new integral formula in the symbolic calculus

Series I: COMMENTATIONES MATHEMATICAE XXV (1985) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PR ACE MATEMATYCZNE XXV (1985)

Piotr Jakôbczak (Krakôw)

On a new integral formula in the symbolic calculus

Abstract. In the note we construct an integral formula for the functional calculus in Banach algebras, with an explicit integral kernel, in a class of domains in C". The formula is based on Henkin’s representation formula for homomorphic functions in convex domains.

1. Introduction. The symbolic calculus in commutative Banach algebras is well known. In one-variable case it is given by the Cauchy integral formula with explicit kernel but for higher dimensions there is no analogous formula in general case. Nevertheless F.-H. Vasilescu found a Martinelli type formula for the functional calculus of commuting system of operators in a complex Hilbert space [5]. Recently, J. Janas [2] found some integral formulas for the functional calculus, based on the representation formulas for holomorphic functions in various kinds of domains in C".

In this note we construct an integral formula for the functional calculus in Banach algebras. This formula is an extension of Henkin representation formula for holomorphic functions in several complex variables [1].

Let D c C" be a bounded convex domain given by a defining function g, i.e., D = {z e C " : g(z) < 0}, where g is of class and convex in an open neighbourhood D of D and (ô g /d ^ , . . . , ôg/ д ^ Ф 0 on 8D = {z eC " : g(z)

= 0}. The usual differential operator d is given by d = ô + d , where

Ô Ô

d — — dzi + ... + - — dzn and a similar formula for c. Write

dz1 dz„

f t ( 0 = ~ ( i ) = D ' e d i cn = ( « - l)!/(2ro7.

Now, following Henkin, we define the (n — 1, n) differential form £?(£) by

0 ( ( ) = C, t ( - 1 Г 1 e , d ) л Д d e j d ) л d i , i=l

where dÇ = d^ л ... л d£n. Let

M d , z ) = t е , т ( , - г , ) , ( i , z ) e S D x D .

i= 1

Denote by A{D) the Banach algebra of all functions continuous on D and holomorphic in D. Then for any z e D and f e A ( D ) we have the following formula of Henkin [1 ]:

( 1 ) / ( 2 ) = J / ( 0 * f ( { , : ) - û ( { ) .

êD

In what follows we shall give an extension of the above formula which will produce the symbolic calculus in Banach algebras. We should stress that our result is contained in that of [2 ] ; however, the proof given in [2 ] is based on a non-trivial result on the existence of the Cauchy-Weil integral formula for the functional calculus [4], while our approach is elementary and uses very little from the theory of Banach algebras.

Some applications of the integral formula derived in this note as well as of other integral formulas for the functional calculus will be given in [2].

2. Let A be a commutative complex Banach algebra with unit e. For a complex number X and х е Л we shall write for simplicity X —x instead of Xe

— x. Consider a system of elements x ls . . . , x ne A . Denote by К the joint spectrum o-(x1, x„) of x l 5 . . . , x n. We can assume without loss of generality that 0 gK . Now take any convex, bounded open set D which contains K . Suppose that D = {z e C n: q (z) < 0}, where q has the properties described above. Note that

i

ми, x)

i= i

is an invertible element of A for every £ e dD. This is clear because

x)) = {X e C : X = £ for some (wl5 . . . , w„)eX } .

i= 1 We claim that the formula

(2) / ( x „ . S f ( i ) M ( t , x ) - " Q ( t ) , f e A ( D )

?D

defines a continuous homomorphism Ф of A(D) into A and for any polynomial p,

Ф(р) = Р (хi, x„) (where p ( x t , . . . , x„) has the usual meaning).

First we introduce a little more notation. Write

£ e ( t ) - t =

t

i = 1

Let Dr = {Rz: z gD] for R > 0. Note that the function qr(Ç) = q(Ç/R)

defines (in the way mentioned above) the domain DR. In what follows MR(Ç, z) and &*(<!;) have the same meaning for DR, as M( £, z) and Q(Ç) with respect to D.

Before we proceed to the proof of the above claim, we have the following result:

Proposition 1. L et P(Ç) = ... £ n, oq + ... +oc„ = |a| = p. F or l ^ 1, R > 0 and z e D R we have

J P(Z)(dQR(Q-ZYMR{Z,z)-''+t>QR(Z)

dD R

= (1 + p/n ) ... (l+ p /(n + l - l ) P { z ) . Pr o o f . Induction on /.

Let / = 1. Differentiating with respect to z; both sides of the equality P ( z ) = j Р ( & м л ( Ь г Г яа А в

dD R

we have

(3) LPP(z) = n j P{t){Q R)j{t)M R{t, z ) ~ ^ " n R{Ç).

ô d r

Hence

P (z )= j P ( ( ) M R( { , z)M R(( , z) - ("+ " Q r ({) SB*

= J i , ( 0 ( a e « ( a - { ) M * « . z ) - <"+ ,' o II( o - mR

- t * j f р т е я Ы о м к а ^ г 1"+1,а к а )

J= 1 ^IDK

= J i >( a № * ( { ) - « ) A f , ( { , z ) - <"+ ,,0 * ( i ) - - P ( z ) ;

s d r n

П \

the last equality follows from (3) and the equality £ ZjD* P(z) = pP{z). This j= i

proves the proposition in the case 1 = 1.

The same method also works for / > 1, so we omit the details. The proof is complete.

Proposition 2. L et P(Ç) = £ l |a| = p. Take a number R = Kj greater than the follow ing two numbers:

2 sup {|£j(£)| •||xk||/(min|dg(i7)*?/i): dD, j , к = 1, . . . , n},

qedD

(sup £ Ы Ш * , 11)/ (т т | ге (1/)-1/|).

ÇeôD i = 1 ^tiefiD

We have for any integer I ^ О

(4) J P ( 0 (8qr( 0 • # M R(£, х)~ (" +г)QR(Ç) = А (1)Р(х1, х„),

ô d r

where >1 (0) = 1 and A (/) = (1 4- р/п) ... (1 + р/{п + / — 1)), / ^ 1.

P r o o f . First of all note that the above choice of R is possible. In fact, for any Ç g ÔD,

R efà?«K ) = (grad e(f), f ) , # 0

(the inner product in jR2"), because D is convex and O eD (by our assumption O eK ). Since ÔD is compact, mm\ÔQ(ti)-rj\ > 0 . Now for Ç edD R we have

riedD

( Î _{i= 1}

t (QM

i)*)'1

= (I WR)e,(i/R)R(UR)-ï.ü/R)Ql(mx,y'

i i

= R (R Y , where V = £ /R ed D .

i i

By our choice of R we can write

я (»/)•»/-£ в М ъ ) 1

R ((l/R )ê Q(r,) r,) £ QT Qi(rj)Xi/Rêg(ri)-r]f

k = 0 i

(R/de№)-() Z Œ . e , № ) * № № ) ■ $ ,

к — 0 i

and the last series is uniformly and unconditionally convergent on DR. But we can rewrite the last series as a power series with respect to X l, . . . , x„

£ су( £ )ху, where

Since

y c U)xy= y s ( е Л У У х Л * ( e . W ) x . f

' , g А ш т - i ) - \ s e m n ) '

#{/> = l ^. = fc} = C t - l ' ) ^ fc"

it is easy to check that the series

Z c y ( 0 x y

is uniformly and unconditionally convergent on 8DR to М л (£, x) 1. Hence the Cauchy product

( X c y(£ )x y)"+I = J X ( £ ) x 7

7 7

is also uniformly and unconditionally convergent on 0DR to MR(£, x )-(n+i)

= L ^r( ^ x)~ 1Y +1 (by the Abel theorem on Cauchy product of unconditionally convergent power series). Thus

J P(t)(dQ R( t ) - t ) 1 M R( t x ) - ^ +l'QR(t)

dDR

S Dr 7

= £ ( 1 Р(М ,(£)(аея(()-(]1оя({))х’

7 dDR

and the last series is unconditionally convergent in A. Similarly for z e { a e C n: |af| ^ ||xf||, i = 1, . . n] = L we have

SBK

= 1 ( 1

7 S Dr

and the series is uniformly convergent on L. By Proposition 1, the left-hand side of thç above equality is equal to A (l)P (z). Therefore,

j

v # a ’

s i M (0 , y = ct.

Consequently,

f P (£) (dgR ( 0 • Çj M R (£, x)~<"+'> Qr ( 0 = A (/) xa = A (0 P (x)

S Dr

and the proof is complete.

We also need the following proposition:

Proposition 3. F o r any positive numbers r, R such that c r ^ , . .. , x„)

= К c= Dr and К c= Dr, and fo r P(Ç) = |a| = p, we have

1

SDr

= (r/R)p+n+l j P(ri)(dQR(ri)-ri)l (Y,(eR)i(ri)((r/ R) r1 i - xi)Y (n+l)QR(rl)’

dDR 1

( = o , 1 , . . .

P ro o f. By the definition of gr, gr(r/R) — gR{Ç) for Ç edD R. We also have (for the same ç)

(É?r)i ((r/R) ç) = (RM{gRh (c), (Qr)ik ((r/R) Z) = (R/r)2 (£),

where {gr)ik = d2 g/c>zi dzk . If </>: R2n~1 =5 A -> dDr denotes a local parametrization of dDr, then ф = (R /r)(p is a local parametrization of dDR.

Let /eCo(<p(d)). Then /(£ ) = x((r/R )Ç )e Со(Ф(Л))- By a direct computation we check that

J

>D,

= 0DR J P(Om)(SeR(()-it((r/R)SeR(i)-i-S(е*)<( « <”+' * («)■ i

Hence applying the standard argument (a partition of unity formed by x) we get the desired equality. The proof is complete.

Now let us proceed to the proof of (2). We have to show that (2) holds for any D = {z e C n: g(z) < 0] for which D з К . Let R 0 = inf (Я: K c DR}.

Define the set

V = \R\ R 0 < R ^ R i and (4) holds for every 1 = 0, 1, ...} ,

where (4) and R 1 are the same as those given in Proposition 2. By Proposition 2, R ^ V We want to prove that V = ( R 0, Æj]. In order to do this it is enough to show that V is open and closed in (R 0, JR J.

(a) V is closed in (R 0, R ^ .

Let Rke V, R k -> R as к

-> oo

Il I Р (0 (ге Як(()-()‘ м „ ((, x)-<"+° o Kl( i ) - т ч

- J Р Ю ( г е * ( а - { ) 'м я ( ё , х ) - " +,>йк (й||^о, as к - oo.

<1D R

By Proposition 3, we have 1 1 - J

fl>Rt !'DR

( R J R ) * + " + ' f a o R)l ( ^ i l - x ,' j\ ' — M K(Ç, x) - ( n+l)

(r j r y *" *'

g с||Л|го,1 № * (0 -« | | ^ ,» о 1 (а о ,)х

~(n + l)

- M R( t , x ) - ( n+l) x sup

tedDR

where c comes from QR(Ç). Now by a straightforward calculation one can show that

SedDRsup (R JR >p+n+l

£ ( е я Ш ( ^ . - - х «

-(и + I)

as

/с —

oo

- М я ( ^ ,х ) - (и+г> - О

(b) V is open in (R 0, Я х].

Let JReK Note that the function 0DR -* 1/||Мя (£, x )_ is continuous and does not vanish on dDR. Choose t, > 0 so small that R 0 < R

— г and for every r e ( R — s, R + s) and Çe ôDr the following inequality holds

i

< inf ( l / i i M n ^ x r 1!!).

«6 dDR

The above inequality enables us to write

£ Ы ,- (£) кp - Z Ы ; (£) *«•

q° / y \k

= 1 Д г е , ( й ^ - - а е11( { ) - { 1 (M R( t , x r T ■

(Here we have used the identity 00

(Я—x ) - ‘ = ( л0 —X) - 1 Y. ( Я о - ^ К А о - х ) - 1]*.)

k = О

Note that the last series is uniformly and unconditionally convergent on 6Dr . Hence

Е ы , ( ‘й (('-/ кк1- * , ) Г 1: Г 1

i

= MR(Ç, x , r (”+l) { t [(1 - r / R ) ( e eR( ( ) - i ) M R((, x ) - , ] ‘ }<" + ') fc= 0

= MR(t , x ) - '" +i» £ ( n+''t ' ; 1) [ ( i- r / R ) ( a e ji( a - « ) M J!« , x ) - 1]', t =o V n + l —1 /

where the last series is also uniformly and unconditionally convergent on 8Dr .

3 — Prace Matematyczne 25.2

Now applying Propositions 2, 3 and the above identity we have

J P(

(

e,(f)-«)'M ,({, x )-‘“+')a r(c)

SDr

= (r/R Y +’ +‘ j

dDR \K i /

= £ ( n+l+,~ l\ i - r/R f j p(o(SeAO-i;)‘*'MR(ï, х Г (”+|+,,а * ({) f = 0 \ n + l- 1 / SDr

= (r/RK+»+' £ (" + ' ^ ; ‘)(l-r / R )'J4(/ + I)R(x) f=0 \ Л+/-1 /

(in the last equality we have used the assumption R e V). Therefore the proof of (b) will be complete if we show the following identity:

(5) (r/RK+-+ , £ ( n+l+'~1) ( l - r / R ) ‘ A(t + l) =A(I), ,= o\ n + l - l J

/ = 0, 1, 2, . . . We consider two cases:

1° 1 = 0.

Then the left-hand side of (5) is equal to

W +' [ l + £ r +' " ‘) ( 1 - r / K H l + p / n ) ... (1 + p / ( n + t - 1))]

t= O ' n ~ 1 /

00 / I I . < \ 00

= (r/R)p+n Z (” ” ~ )(l-r/ R )' = ( r/R)<’+”( X ( { R -r y R f),+p

t= o v П + Р - 1 / f c =о

= (г/Я)р+п(Я/г)р+п = 1 = A(0) and the proof of 1° is complete.

2° 1.

First, note that (n+l+t ^ A H + t) = A{l)(n+l+P+t Hence the left- V n + l - l / V n + l+ p -l /

hand side of (5) is equal to

(Г/ЛГ"Л(/) z ("+l+. P+‘~ l )((R -r )/x y ,= 0 \ n + l + p - l Г

= (r/R)p+nA ( l ) ( f À ({R -r )/r)y +l+p = A(l)(r/R)n+p+l(R/r)n+p+l = A{1).

t= о

This completes the proof of (b).

In this way we have proved that V = (R 0, f^ ].

Since the mapping Ф: Л(£)л)э/-> f /(£)М Я(£, х)- "&д(£)еЛ is linear,

ôdr

for any polynomial p we have

p ( x ) = J p(!;)M R(Ç ,x )-" Q R(t), R > R 0.

SDr

It is also clear that Ф is continuous on A(DR). Since Ф is multiplicative on polynomials and every f e A ( D R) can be uniformly approximated by polynomials on DR, so Ф must be multiplicative on A(DR).

Summing up, we have proved the following

Th e o r e m 1. Let A be a com plex commutative Banach algebra with unit e.

F or any R > 0 such that c r ^ , x„) c= DR the mapping Ф given by A(D R) 3 f-> J f( Ç ) M R(Ç ,x )-" Q R( t ) e A

dDR

has the follow ing properties:

(a) Ф(р) = p(x) fo r any polynomial p, (b) Ф is a homomorphism o f A(D R) into A, (c) i f f k = t f, then Ф (Л )-> Ф (Л in A.

dr

It turns out that Theorem 1 has an extension to more general domains D =) x„). Namely, following Range and Siu [3 ] we assume that D is a bounded domain in C", for which there exist a neighbourhood U of ÔD, an open covering {£/,}*= i of U and functions Q jeC 2(Uj, R), j = 1, . . . , к such that

(a ) D n U = { z e U : z $ U j or Qj(z) < 0, j — 1, . .. , k },

' i

(b) for every 1 < i1 < ... < ц ^ к and 26 П Uiv the vectors

v= 1

gradg,- (z), . . . , gradg,- (z) are linearly independent over R. In particular, if

1 i 1

l > 2n, then П Ui — 0 .

v= 1

Let I = (il5 . .. , I,), where i1} . . . , ц е {1, . . . , к}. Define Sj

=

) = 0,

iel

Assume that D has the natural orientation (induced from C"). We choose the orientation of Sj which is skew-symmetric with respect to the indices of / and for which the following conditions are satisfied:

(i) ÔD = ( j

j = i

(ii) = и Sj

j= i 1

In what follows we shall assume that all functions gt are convex. Define (а)Д0 = ~ ( i ) , м ,( { , x) = t ( e M O ( i j - x j ) ,

d 4 j = 1

j = 1, n, i = 1, k,

iQ M t)'

(.а Ш ,

Fix I = (i1, . . . , ij), where 1 ^ < ... < i, ^ к and 1 < / ^ n. Let Г = { J

= (/i> Л е Д v = 1, n — /]. For any J<=T we put

= det(Lf l (£), Lfl(£), dLjn_ ^ ) ) d ^ Л . . . Л С / (ftjM É ), . . . . (а Л Ю , 2 ( ^ ( 0 , . .. .

= det I i ) л dÇ.

. . . . ( & ,Ш , ^ ( ^ Ш , . . . . ^ п_ гш ,

The above determinant is defined as usually, but the multiplication is replaced by the exterior multiplication (for example ^(^71)i(^) л d(gj2)2 (£) instead of c'(gj1)i(Ç )‘ d(gJ )2 (Ç))- Since the exterior multiplication is not commutative, so the determinant can be different from zero, even if there are the same columns in the above matrix. T h e r e f o r e i s a differential form of (n, n — l) type.

Now applying the results of [3] and repeating the reasoning given in the proof of Theorem 1 we have the following

Th e o r e m 2. L et D, gh i = 1, . .. , k, be as before. Take x 1? . .. , x ne A such that a ( x lf x n) c D. Then fo r every I = ( i l5 . . . , i,), 1 ^ ij < ... < it ^ k, 1 ^ ^ n and every J e T there exist constants y/5 a u such that fo r every f e A ( D )

(6) x„) = E Г/ Z a ,j J / ( £ ) [ M ( l « , x ) . .. x )x

\I\^n Jel' Sf

where |/| = / (if к < n, then £ is understood as £ ).

î^m^n к и к *

For example, if к = 2, then (6) has the following form:

f ( x u . .. , x„)

= сй[ | / ( а м 1( ^ ,х ) - " t ( - l y - ' t e i M f l л Л 4 Ш а<« +

Sj j = 1

+ j X)-" f ( - l f ' f e y i ) л Д 3 (e 2m * < K +

s 2 j - 1

+ У12 £ ai l w J / ( £ ) [ A / , ( f , x ) M2( { , x ) ] - ‘ x

J -U \ ,—,jn- 2) S(12)

J ve { l , 2 }

n — 2

X [ П 4 v ’ x)] Ц 1

(£) •

v= 1

References

[1] G. M. H enkin, Integral representations o f functions holomorphic in strictly pseudoconvex domains and some applications, Mat. Sb. 78 (1969), 611-632 (in Russian).

[2] J. Ja n a s , On integral formulas and their applications, preprint.

[3] R. M. R ange, Y.-T. Siu, Uniform estimates fo r the v-equation on domains with piecewise smooth strictly pseudoconvex boundaries, Math. Ann. 206 (1973), 325-354.

[4] J. L. T a y lo r, The analytic functional calculus fo r several commuting operators, Acta Math.

125 (1970), 1-38.

[5] F. H. V a sile scu , A Martinelli type formula fo r the analytic functional calculus, Rev.

Roumanie Math. Pures Appl. 23 (1978), 1587-1605.