Báo cáo nghiên cứu khoa học: "Analytic perturbations of the $ar partial $-operator and integral representation formulas in Hilbert spaces " pps

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1 (1979), 187-205

ANALYTIC PERTURBATIONS OF THE 0-OPERATOR AND INTEGRAL REPRESENTATION FORMULAS

IN HILBERT SPACES F.-H VASILESCU

1 INTRODUCTION

In this paper we present the construction of some operator-valued kernels which occur naturally in the study of certain integral representation formulas, in particular in the analytic functional calculus for several commuting operators in Hilbert spaces These integral kernels are obtained in connection with the analytic perturbations of a specific type of the d-operator, when @ is regarded as a closed operator on Hilbert spaces of square integrable vector-valued exterior forms

Let H be a complex Hilbert space and @(H)((H)) the set of all densely defined closed (bounded) operators, acting in H For any Te ằ(H) we denote by A(T), A(T), H#(T) the domain of definition, the range and the kernel of 7, respectively

In what follows we shall deal mainly with operators Te @(H) having the property A(T) c #(T), ie., roughly speaking, with operators T satisfying T* = 0 Such an operator T' will be called exact when one actually has &(T) = #(T) The exactness of an operator T Ạ @(A) with A(T) Ủ X(T) is equivalent to the invertibility in L(H) of the operator T 4- T*, where T* denotes the adjoint of 7; this is a simple and useful criterion from which some of the main results of this paper will be derived, Let us consider a finite system of indeterminates o = {;, .,0,) The exterior algebra over the complex field C generated by o,, ., 0, will be denoted by Afo]- For any integer p, 0 < p <n, we denote by A?[o] the space of all homogeneous exte- rior forms of degree p in a1, ., 0, The space A[o] has a natural structure of Hilbert space in which the elements

ử, NA su AG; J (<j< <<m;p=l, ,n)

as well as 1 eC = A%a] form an orthogonal basis (the symbol ỘỔ AỖ stands for the exterior product)

Let us define the operators

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188 F.-H VASILESCU

Then the adjoints of the operators (1.1) are given by the formula

(1.2) SHG +a A )=đ, G=l, ,n),

where ằj -+ a; ÉjẼ is the canonical decomposition of an arbitrary element Ạ Ạ A[o], with ằ; and ằjỖ not containing ằ; Note the anticommutations relations

S;S, + S,S; = 0

(1.3) (j,k =1, ,n),

S,)Sé + SES; = x

where é;, is the Kronecker symbol, which can be readily obtained from (1.1) and

(1.2)

For an arbitrary complex linear space L we denote by A[o, L] the tensor pro-

duct L ệ Afo] If 2 is any endomorphism of L then the action of 2 is extended on

Ala, L} by the endomorphism 1 @ | We identify these endomorphisms and keep the notation 4 for both of them Analogously, if @ is any endomorphism of A[o] then the endomorphism 1 @ 8, acting on A[o, LJ, will be also denoted by ử

Any commuting system of endomorphisms ụ = (a, .,4,) acting on L will be associated with the endomorphism 6, on A[o, L], defined by the relation

(1.4) 6,6 = (aS, + +4,8,)E (ẠẠ Ae, L))

From (1.3) we have that (6,)? = 0

Assume now that L is a Hilbert space H Then A[a, H]is also a Hilbert space The action of any Te Ạ(H) will be extended by T @ 1, denoted simply by 7, defined on ZT) ệ Ale] = Alo, B(T)} Clearly, for any endomorphism @ of Ala] we have ĐT c T0 (for Tị, 7T; in @(H) the notation T, < T, means that 7; is an extension of T))

de: @ be an open set in CỢ and Cệ(O, H) (A(Ó, H)) the set of all /-valued indefinitely differentiable (analytic) functions on Q Consider a commuting system

% = (4, .,0,) in A(Q, ặ(H)), i.e a system of operator-valued analytic functions such o(z)a,(w) = o,(w)o,(z) for any j,k = 1, ., n and z, w in Q The corresponding endomorphism (1.4) for L = CỎ(Q, H) will be then given by

(1.5) ỗẤÈ() = ((Z)Ế +- + %(Z)S,)ỌỂ), (zeÓ),

where Ạ Ạ A[a, Cồ(Q, H)] We can consider also the usual ử-operator

(1.6) a= Oo đZ;+ +Ở đzZẤ,

Oz, Zm

acting in the space A[dz, C*(Q, H)], where z = (zạ, ,ZzẤ)Ạ @ are the complex

coordinates and đz = (dZ;, , đZẤ) ¡is the corresponding system of differentials

Then the endomorphism 6, + 0 acts in the space A[(ụ, đZ), c~(Q, H)}, with (o, dz) =

(G,, +,6,, 2,, ,dz,,), and has the property (6, + 0)? = 0, since 5,0 =Ở06,

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integrable exterior forms Unlike in some works dealing with harmonic forms on strongly pseudoconvex manifolds [3], [1], or in the Hodge theory [8], we shall try to emphasize the role played by 7 + 7* rather than of TT* + T*7, where Te ằ(H) is an operator with the property @(T) < X(T) Indeed, it is such an operator which leads us to a class of natural kernels yielding integral representations formulas in Hilbert spaces (see also [6]) Among some applications, we show that the usual multiplicativity of the analytic functional calculus for commuting systems of operators follows from a more general characteristic trait, namely from a property of module homomorphism over the algebra of complex-valued analytic functions Let us mention that the results of this paper have been partially announced in [7]

2 THE d-OPERATOR IN HILBERT SPACES

From now on H will be a fixed complex Hilbert space Let Q be an open relatively compact subset of CỢ and L(Q) the usual Hilbert space of all (classes of) complex-valued square integrable functions on 9, with respect to the Lebesgue measure Let us denote by Hg the completion ặ7(2Q) @ H of the tensor product L*(Q) ệ H with respect to the canonical hilbertian norm In other words, Hg is the space of all (classes of) H-valued functions, strongly measurable on 2 and whose norm is a square integrable function [5] We shall use also the notation Cg for Lồ(Q)

Let us fix a system of indeterminates Ạ = (ằ,, ., ằ,,) and define the operator din A[f, Hy] As in the scalar case [2], we shall use the way of the theory of distri- butions Every element & Ạ A[Z, Hg] can be associated with a A[Z, H]-valued distri- bution v; by the formula

(2.1) vp) = \ ọ(z)ặ) d2(z), (p C#(ệ)),

where dÀẢ ¡is the Lebesgue measure and C?ồ(ử) is the usual subspace of CỎ(Q) (= Cồ(Ó, C) of all functions having compact support We may therefore consider the areolar derivatives 0v,/0z; as well as the operations CAV = VERE (j= 1, , m) In this way the formula

= 0 5 a,

due =(ỞỘ-B ve ( az, oy + 4 4 + az | A Ve

m

makes sense and defines the operator 0 within the theory of distributions

We denote by BQ) c 4[ễ, Hạ] the set of those ằ Ạ A[Z, Hy] such that there

exists an 7 Ạ A[f, Ho] satisfying Ov, = v,; we set 0Ạ =n In other words we have

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190 F.-H VASILESCU

for any g Ạ Cf(Q) The formula (2.2) shows that the operator 0 is a weak extension

of the operator (1.6) (We prefer to use the system [= (f,, ,ằ,,) instead of

dz = (dz,, .,dz,,) in order to stress the independence of the former on the points in Q.) As in the scalar case, the operator @ is closed and densely defined In fact, if Ạ is in A[Z, Cồ(Q, H)] and both ằ and dé belong to Ate, Hl, where 0é is defined as in (1.6) with an obvious identification, then Ạ Ạ G(@) and Ạ satisfies also (2.2) A nother useful remark is that if Ừ ằ A[f, Cồ(Q, Y(H))] and y with its derivatives are bounded on @ then for any Ạ Ạ BA) we have also pa ẠG(0) and O(yA 2) can be

calculated according to the rules of the exterior derivative Indeed, the formula (2.2) is still valid for @ Ạ C9(Q, Y(A)); this last assertion follows from the density

of Co(2)@ L(A) in CH(Q, L(A) [5]

The most important feature of the operator ô is that ử(2) c # (0), as one can see from the formula (2.2) Therefore 0* has a similar property Let us denote

by 0, the ỘscalarỢ operator @, ie the operator 0 obtained for H = C We shall see that 0 is the closure of 0,@1, defined on ZO) @H

2.1 LEMMA For any ẠẠ (0) there is a sequence ẽ,e 2(ô,) @ H such that

E, + Ạ and (0,@ 1)é; > 0Ạ as j > co, in All, Hol-

Analogously, if & Ạ G(A*) then there is a sequence Ọ;c2(0*)@H such that

ễ; Ở Ọ and (0* @1) E; > 0*E asj + 00, in A[l, Hol

Proof Let us fix Ạ Ạ 9A) Since the coefficients of Ạ and 0é are strongly measu-

rable functions and we are interested to approximate their values with elements of H, with no loss of generality we may suppose that His separable Assume that{e,}2 ,

is an orthonormal basis of H Let us represent Ạ = 3 Ữ;Ấ, where J = (i,, , 7,)

Lf

is an arbitrary multi-index with | <i, < <i, <m and fp= Ci, A AC (The symbol ỔỔ@ỖỖ will be generally omitted when representing exterior forms; it will be used only to stress the aspect of certain forms.) Analogously, n = đệ

will be written as 7 = > ny Let us define the operators

(2.3) mz) = Ừ <7(Z), erent, (2 Qk = 1,2,3, ),

for any = Myf, e Alf, Hq], where the scalar product is in H According to the

1T

definitions (2.1) and (2.2) we can write Vign (Q) = \ ụ(2) Đ (n2), eÈ su, đA(z) =

I

=u (ầ ((eene ait) tr}= Ở af & ({Be@ EGE) 4 t,Ì=

f

~ x 9 2 = (2.4) d4Ể) = ÔụẤẤ:(@),

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JỞẲỂỦ

j j

Let us define now ằ; = Yas and 4; = UE We have lim ằ,(z)=ằ(),

kei k=l

lim y,(z) = n(z) almost everywhere and jỌ;(z)j < ]Ư(2), lin(z)' <'n(z)|, by the Bessel joe

inequalities Therefore the Lebesgue theorem of dominated convergence implies that Ọ, Ạ and n, = 0ằ, > n = 0ằ as j Ở 00, in Alt, Ho)

Next we show that ầ <é,(+), e,>Ạ, Ạ Z(O,) for any natural k Indeed, if we

7 define the operator

wi ầ2707) = Yc e,SỌ; Ạ A[E, C]

I I

for any ` x,ỗ¡c A[Ế, H], then we have for cach ọ Ạ C$*(Q) i

\ 9) ầ; <n), ed) dA) =

i

=wy y ậ leone) ax) 0 = I

= ỞỪ ( x Đế (Ez) daz) Er ) =

1 i

= Ở[do@) 0 Yi <E@), 0) CG), I

which proves that 0,ầ, <2,(z), e.>01 = Y <ni(z), ex>f; Consequently the elements

i 1

ế; constructed above belong to 2(0,)@H, which finishes the proof of the first part

of Lemma 2.1

Consider now éẠQ(0*) and notice that we may still suppose H separable

and that {e,'@., is an orthonormal basis in H It is clear that the operator u, given

3

by (2.3) is self-adjoint (in fact, , isa self-adjoint projection), therefore ằ; = ầ, 4 is

kel

an element of 2(ử*) since

Ở j Ở

<ế;, 0y) = (y uy! tế, rỒ, k=1

for any y Ạ (A), on account of the first part of the proof The same argument using the Lebesgue theorem of dominated convergence shows that ằ; + ằ and 0%, = Oe

as j > oo, in All, Hol)

Take 0= JY O,f,Ạ Q(0,) arbitrary Then from the formula (2.2) we infer J

easily that 9@x eA) for any xe H, therefore assuming Ạ = VY ằ,f, and OE =

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192 F.-H VASILESCU

= }) n,ằ, we shall have

Ạ3 Ỉ<66), e2 bu 9, 38,0} =

=ẹ; E1(#), &> e675 98,80) = = <nil*), > ey Ấn, Y0 @exbs> = =h ny(*), n> Cy, 3,6;

hence ầ} <Z,(*), SỌ, e 2(0#) In this way ej c2(0*)@H for any j, and the proof 1

is complete,

2.2 THEOREM The operator 0 is the closure of the operator ô,@1 Analogously, the operator 0% is the closure of the operator 0* @1

Proof As we have already noticed in the previous proof, 9(0,)@H < 20); by Lemma 2.! we obtain that ử is precisely the closure of 0,@1

Concerning the second assertion we have only to prove that Z@*)@H <

c 2(*) Indeed, if (Y 6,ặ)@x is in GO*)@H then for any (ầ é,f,)@y in

7 J

BO) @H we have

Ề(*0;)@x, ay Ef)@y =

Ề(đ* 3.0,,)@*x, (40) 7

and approximating any Ế e 2(2) with elements from B(0,)ệ Hin the sense of Lemma 2.1 we obtain the desired conclusion

Theorem 2.2 suggests that many significant properties of the operator a, can be formulated and proved for the operator 0 too As a sample, we shall show that if Q is a strongly pseudoconvex domain in CỢ (in the sense of [l]) then the range of 0 in A[ặ, Ho] is closed For the operator ở, such a result is a consequence of the deep theory concerning the 0-Neumann problem, developed by J.J Kohn [3], [I] We need some auxiliary results, which can be formulated in a more general context

Let us fix an operator Te @(H) such that A(T) ằ #(T) 2.3 Lemma The operator L = TT* + T*T is self-adjoint

Proof The result is given in [3, Prop 2.3], so that we only sketch its proof It is enough to show the relation

(L+ 1p* = (1+ T*T)?* + i + TT*)7 ++I,

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Proof Obviously, B* > B Let us show that B is closed For, take x,ằZ(B) = = &(T) 2 &(T*) such that x, > x and Bx, > y as k > co We can write x, =

= xXƯ + xy, with x, Ạ X(T) and xj! Ạ 2(7*) Then x, > xỖ e X(T), x > x" Ạ A(T*); Tx; > y' Ạ X(T), T*#x, ể y'e 2(T*) and y = y' + y'' = Tx'' + T*x'=Bx, hence

B is closed

Assume now that x) ằG(B*) is such that the pair {x , B*x9} is orthogonal in H @ H on.the graph of B Then we have <x, x> + <(B*xy, Bx> = 0 for any xẠG(B), whence B*xyằG(B*) and B**x, = Ở x) By Lemma 2.3 we obtain

(1 + L)xy = 0, thus x) = 0 Since B is closed we must have B = B* 2.5 COROLLARY We have the orthogonal decomposition

(2.4) H = &(T) @ A T*) @X(T+ T*)

Proof The equality (2.4) follows from the relation

Q3) ãữ + 79 = 4ữ) @ 29,

whose proof is straightforward

2.6 LEMMA The space &(T) is closed if an only if the space @(T + T*) is

closed ồ

Proof The assertion is a consequence of the equality (2.5)

Let us return to the operator 0 when acting in strongly pseudoconvex domains 2.7 THEOREM Assume that Q < CỎ is strongly pseudoconvex Then (0)

is closed in A[C, Hg)

Proof Let us consider the self-adjoint operator L = 0,0* + a*9, It is known that A@(L) is closed in A[Z, Co] [3], [1] Therefore we can write L = L, @ 0 with respect to the decomposition A[Ạ,Cp] = A(L) ệ #(L), and Ly is self-adjoint and has a bounded inverse on &(L) Note the identification

(2.6) AlỖ, Hạ] = (2) @ H) @ (Z(L) @ H)

The operator Lằ!@1 has a bounded self-adjoint extension L>1@ 1 on AL) @ H, which must be injective since the range of Lo! @1 is dense in #(L) @ H Then the operator Ly) @ | has a closed extension L, @ 1, whose inverse is Lo! @ 1 In this way the operator L @ 1 has a closed extension L @1= (Lạ @ I)@0on A[Ế, Hạ] by (2.6), and the range of L @1 is closed Obviously, L @ 1 is also self- adjoint

Let us prove now that L @ 1 is exactly the operator 60* + ô*ô Indeed, if Éc ử(L) then by Theorem 2.2 we have Ọ @ xe 2(08* -+- 8*ử) for any xeH, there-

fore dd* + 0*0 L@I Since both L @ I and ửđử* @ ô*ô are self-adjoint (the

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194 F.-H, VASILESCU

if zero is an isolated point of its spectrum, we infer that the range of 0 + 0* is closed in A[Ạ, Họ], therefore the range of @ is closed, by Lemma 2.6

Since for Q 4 @ and AH # {0} the operator @ cannot be exact, Theorem 2.7 is the best information about @ on this line

3 ANALYTIC PERTURBATIONS OF 2

Let U be an arbitrary open set in CỢ and a = (a, ., %,) a commuting system in A(U, #(A)) We denote by *,(a, H) the set ofall points ze U such that the system a(z) == (a,(z), ., %,(Z)) is singular as a commuting system of linear operators {4]- The set (a, H) is closed in U (it may be either empty or equal to U in certain

cases), therefore the set U\7y (x, H) is open |4}, [6] We associate the system

a = (%, ,%,) with the system of indeterminates o = (0,, ,ằ,) The system

Ễ = (Ế¡, -; 6Ấ) will be associated, in the sense of the previous section, with the operator 0 It is known that for zằ U\P,y(a, H) the operator 5,,, + 5%,,, where

5,(.)is given by (1.4), has a bounded inverse on Alo, H), therefore (d,(.) + O%2))"* is an element of Cồ(Q, L(Alo, H])), for any open Q c UNI, (a, A) {[6]; see also Lemma 3.1 below) When Q is an open relatively compact subset of U\ Py (x, H) (ie., the closure of Q is also contained in U\S, (a, H)) then we may consider the operator 6, + @, acting in Al(c, 0, Hg], where 5, is given by (1.5) When defining the operator 6, ổ we take into account the following canonical identifications:

Al(e, ẹ), Hol = Alo, ALl, Hall = AG, Alo, Hal] = = Alo, All, Hg] = ALE, Alo, Hol

We start with the unbounded variant of a result in [6], stated in the general case

3.1 LemMa Assume that Te @(H) has the property @(T) < ⁄(T) Then T is exact if and only if T +- T* has a bounded inverse on H

Proof Wf &T)= X(T) then, by Lemma 2.6, (7+ T*) is closed If xeX (T+ T*), as Tx and T*x are orthogonal, we have Tx = 0 = T*x But x = Ty, therefore 7*Ty = 0, whence x = 0 In this way (T+ T*)! exists and is everywhere defined, hence (T+ T*)1e (A)

Conversely, if (T+ T*)1+ằ Y(H) then, by Lemma 2.6, @(T) is closed and H= R(T) ẹ A(T*), from (2.4) Consequently, #(T) = &(T)

3.2 COROLLARY If T is exact then we have the relations

(T + T*) 17x = T*(T + T*) 1x, (xeử(T)),

(ỂT+T*1T*y= TỰ + Ty, (ye WT)

Proof Wf ve A(T) then v= Tv, wÍth pụcÝ(T*), hence (7+ T*) 1đ = = v, Ạ #(T*) This means that (T + T*)14(T) < (T*) Analogously, we have

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Take now xe Q(T) Then x = Xp + Xy, Xo H(T) and x,eX(T*) We have then

(T + T*)?ồTx = (T ~ T*)1Tx, = (T+ T*)(T + T*)x, = = (T + T*\T + T*) x, = T*(7 + T*) '!xị = TệỂT Ở Tệ) x The second relation can be obtained in a similar way

Let us return to the case specified at the beginning of this section

3.3 LEMMA Consider an open set U < CỢỎ and a commuting system 2 = = (4, -, &) CC Á(U, Y(A)) If Q is any open relatively compact subset in U\ (x, H) then the operator D, = 5, + 0 is exact in A[(ụ, ẹ), Hol

Proof We use an argument similar to that of Theorem 3.1 from [6], with some modifications due to the unboundedness of 0

Consider 7 ẠD(D,) such that D,y = 0 With no loss of generality we may

suppose that y is homogeneous of degree p <n +m in Oy, + +5 Gq Ên, ., Cực Then we represent = fạ -È ỉy + + 4,, where n, is of degree j in G, , Cm and of degree p Ở jin o,, .,6,; moreover, by (2.2), each y, is in (0) We shall be looking for a solution Ạ of the equation D,é = yn, where Ạ = 9 + ằ, + + Spa Ạ; being of degree j in f,,.-.,ằ, and of degree p Ởj Ở 1 in o,, .,6, By identi- fying the forms of the same type we obtain the system of equations

5450 = No

46; + 064=n; G= 1, -.-a Ở Ì),

La = Tp

with the conditions

Ế Ấ1a =0

ôẤn; Ôn,-ạ Ở= 0 (j=1, ,p) ôn, = 0

Let us define QO(a(z)) = (6,.) + Ox.) 4 (z Ạ Q) and note that Q(a(z)) and its

derivatives are bounded on Q, according to the choice of Q Define also &,(z

= Q(a(2))no(z) By Corollary 3.2, applied to 6,(, we have ằ,(z)ằ.#(6x.)), for every

zẠQ Moreover, as ny Ạ ZA) we have also Ạ,< D(A) and 506) = Ở05,ặ5 = ỞOny =

= ôm, whence 6,(n, ~0é)) = 0 Define then ặ&,(z) = O(a(z))(n,(z) Ở 0&)(z)) Ạ

EH (Oxo) (ZẠ Q), hence 6.6, =m Ở0Ạ We have also 5,ặ, Ạ G6), therefore 5,(Y Ở 0Ạ,) = Syn, + 08,, = 0, which allows the continuation of the procedure

One has, in general, ỉ(z) Ở 0Ế; ;(2)e2Ý(6Ấ;), hence E(z) = OCa(z))\(nz) Ở

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196 F.-H VASILESCU

3.4, COROLLARY With the conditions of Lema 3.3, if n Ạ A*{(c, 0), Hg] has the

property Dy =0 in Q then

Bae) = % Ế Ở1*0GG)(ử0ỦG)Ữ*n,s@), Ủ9) J=0k=0

is a solution of the equation D,é = y in Q, where O(a(z)) = (64.) + 62,,) 1 and nj

is the part of n of degree j in%,, -,Ạ,, Moreover, By Ạ A?Ộ {(a, f), Hol

Proof The solution B,y of the equation D,é = y is the explicit form of the

solution constructed in the previous lemma

Let us consider the differential operator

(3.1) ( 0 Zt tac Z8),

acting in A[Ế, Cệ(Ó, H)], where @ c CỢ and Z# corresponds to Ễ; by the relation

(1.2) It is easily seen that (3.1) is the formal adjoint of the operator 0 It is also clear that the operator (3.1) has an extension, in the theory of distributions sense, in the space A[?, H,] Furthermore, we have the following

3.5 LEMMA Assume that Qa CỎ is open and relatively compact I, ny Ạ All, Ho) has compact support and n Ạ G(O*) then O*4 = $y

Proof Variants of this result for the scalar case can be found in [1] and [2] As we need parts of the argument in the sequel, we shall give a complete proof Let us take x Ạ C&(CỢ) such that supp x = {z; ||zl| < 1} (where ỘỔsuppỢỖ stands for the support), y(z) = x(Ởz), 7 2 0 and \ y(z) dA(z) = 1 Define then y,(z) =

=e ?my(z/e), for any ằ > 0 Let us denote by 4, the convolution product y,#y; analogously, y, = 7,#y, where y = 0*y Note also that y is still with compact support Indeed, if w Ạ CSồ(Q) is arbitrary then a direct calculation from the definition of 0*

shows that Ở ay Ở Ở Z#⁄+ -+ se Ban + worn 2) 2m - a @.2) 2*(jnq)= Ở ( a

In particular, if y = 1 in a neighbourhood of supp 7, we obtain that supp y has to be compact

As in the scalar case [2], we have n, > and y, > y in A[{, Hạ], as e Ở 0

We shall prove that 0*y, = y, Indeed, according to the properties of y,, we can

write

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for any 0e 26); the equality ô(x,+8) = 7,200 is true in a neighbourhood of supp 7, on account of the relation (2.2) If the coefficients of @ are in C*(Q, H) then

_ m 9 Ở `

dn, 0 = Ở (.W -Ở-Z#n, 0,

41, 9 CS 02; 1 /)

therefore 0*n, = Ở ` (a/z,)Z}n, because of the density of such 6Ỗs in A[f,Ho]

j=l

Finally, if @ Ạ Chồ(Q) is arbitrary then, with the notation (2.1), one has that

U,(p) = \ p(z)y(z)dA(z) = lim \ @() ử*n,(z) daz) =

= lim | ( Ế e0 j=l Tj 23) nae) dA(z) = [- Ế +3; D, jo,

i.e 0*7 = Ôn in the theory of distributions sense

3.6 LEMMA With the conditions of the previous lemma, if yn is also in B(0)

and n = Yi nyt; then On,/0z;ằẠ H for any I, j and ` 7

i One? = |lani2 2

yy {' ae We) = Non? + 8|,

IT j=l, OZ; j;

Proof Assume first that the coefficients of y are functions from C*(Q, H), with compact support Then, by the relations (1.3), we can write in A[f, Hy]

lửn|? + |9nl|2 = <(#đ + 98)n, n> =

m C2 - -

= Ở Ừ 02, az, Ừ Hiện x 1 ò Ỉ Ệ =-

=x l j/=IL S| - i dA(z) l2; |

Take now zƯ as in the previous lemma Then for yn, = 7,*y we have 4,7 n, ôn, = z,*Ôn Ở ôn and On, = x,*0nỞ>0n as e > 0, in A[Z, Hg] By the preceding calcula-

tion we obtain that 0n,/0z; are elements of Hg and the equality still holds

We can prove now the main result of this section By a smooth form we mean

any element Ọ e A[(ơ, Ữ), Cệ(G, H)] with é, dé in A[(ụ, 2), Hol

3.7 THEOREM Consider an open set Ua CỎ and a commuting system

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198 F.-H VASILESCU

Proof The invertibility of D, Ở Dệ follows from Lemmas 3.1 and 3.3 The second assertion is a reflection of the regularity of the solutions of the elliptic diffe- rential operators Let us reduce our problem to a manageable case

Assume first that Ạ Ạ A[(o, 9, Hp] is a smooth form such that D,ằ = 0 By

Corollary 3.2 we have that D,R,ằ = Ạ and D?FR,é = 0 It will be enough to prove that for any We C&(Q) the form 7 = WR,Ạé is smooth Since R,ằ Ạ G(A*), we have also 9 Ạ Z(d*), by the formula (3.2) From the equalities satisfied by R,é we obtain that

On = OW A RE + WE Ở 5,R, 6), ô*q = (Uy Ở Jđ*)R,ặ,

where U, = Ở ((Op/0z,)Z# + + (0W/0z,,)Z%) By Lemmas 3.5 and 3.6 we

obtain that if 7 = ầ) n,ằ,, with n, ằ Alo, H]g, then On,/0z, are still in Alo, H]p,

for any J and j Analogously, On;/0z; are in Alo, H]g (note that Lemma 3.6 can be stated with 0n,/0z; instead of 07,/0z;, with a similar proof) We can apply now an induction argument (see [2], Th 4.2.5(b)) in order to show that the coefficients , belong to any Sobolev space W? (Q, A[o, H])(q20) of those A[o, H]-valued functions on 2, whose derivatives up to the order q are square integrable, therefore 7, are smooth functions, by the Sobolev lemma

Let us obtain the assertion in its full generality Consider a smooth form éẠ A[(c, 2, He) By Corollary 3.2 and the previous case, ằ, = DầR,é = R,D,é is smooth, hence Ạ, = D,R,ằ = Ạ Ở é, is also smooth We have therefore the rela- tions OR,é = é Ở 6,R,é and O*R,Ạ = Ạ, Ở oF R,é It is clear that the preceding procedure can be again applied to WR,é for every Ạ Cậồ(Q), which finishes the proof 3.8 COROLLARY With the conditions of the previous theorem we have the equa- lities

R(H(D,) 0 AMG, 8), Hal) = (D3) 0 APG, ặ), Hạ]

and

R(# (Dz?) 1 AMG, Ô, Hạ) = #(Đ,) n_4?*1(ụ, Ô, Họ],

for any integer p, O<p<n-+m, where AP(c,f), Ho] is zero for p= Ở1 and Pp=m-+n~Ở Ì

Proof The equalities follow from Corollary 3.2 and from the structure of the

operator D,, mapping AỖ((a, %), Ha] into A?*{(a, 0), Hol

Ổ4 SOME INTEGRAL FORMULAS

Let U be any open subset of CỢ ando=(o, -,0,), Ạ=(&%, -5 Ga) systems of indeterminates Let us denote by H/fằ the space of all (classes of) strongly mea-

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&c A[(ơ, Ế), HỊ*] then we deựne ifs integral on a bounded Borel set McU in the

following way: Denote by Ạ,, the part of Ạ of degree min &, ,Ạ, and by Ạ, the form obtained from Ạ,, by substituting Ạ,, ., Ạ,, with dz,, ., dz,,, respectively

Then we put, by definition

(4.1) \ &(z) A dz, A A đzẤ Ở Ẳ Ez) a dz A A dz

M M

C early, the right side of (4.1) make sense and it is an element of A[o, H]

The integral (4.1) does not suffice for our purpose We need a more complicated concept, valid for some smooth surfaces of real dimension 2m Ở 1 First of all

notice that the operator 9 makes sense as a closed operator in A[f, H/e*] (hence also in A[(ơ, Ế), Hồ]), when A[Ế, 19ồ] is endowed with its natural topology of Fréchet-

Hilbert space, giving 0 a similar meaning with that from the second section Then for any open relatively compact 4 < U, whose boundary ặ is a smooth surface,

and for any Ạ Ạ ZA) in A[(o, 0), HỊ??] we define

(4.2) \ E(z) A dz A A dz, =| ôặỂ@) A dãy A A đzẤ, 4

where the right side is given by (4.1) Plainly, the formula (4.2) is suggested by the Stokes formula In particular, if ẠẠ (0) has compact support and we denote by Ạ, the convolution product y,*, where y, has been defined in the proof of Lemma 3.5, then Ạ, > ế and de, Ở 0& as e > 0, therefore

( E(z) A dz, A A đzẤ = lim ÉÁ(2) A đã A A đzạ >0

Consequently, if 7 = dé and the support of & is contained in A then

(4.3) \ n(z) A dz, A A din =| n(z) A dz A A dZ_ =0

A U

The definition (4.2) makes sense also for forms defined only in neighbourhoods of Yin U Indeed, if Q > FJ is an open relatively compact subset of U and @ Ạ Cf(Q) is equal to 1 in a neighbourhood of ặ then for any & Ạ Bd) in A[(o, 2), Hg] we have wt Ạ ZO) in A[(o, 0), H}ồ] by natural extension, hence we may define

(4.2)* \ E(z) A dz, A A dzp =( dpe(z) A dzy A A dZp

2 4

By the above remarks, the formula (4.2)* does not depend on the particular choice

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200 F.-H VASILESCU

Suppose now that x = (%4, ., #Ấ) c Á(U, (H)) is a commuting system such that Y,.(x, H) is compact in U Then the operator B,, given by Corollary 3.4, maps the kernel Z{(ụ, ỷ), HỊP*] of 6, + 0, when acting in the space A[(o, f), H/9*], into the

space A[(o, ặ), HI], where V = U\Y,(2, H) From now on we shall denote by P- the projection of A[(c, f), H}ồ] onto A[ễ, HỊP]; note that P20 c 9P, Let us

also assume that # u(@, H) < A, with A as above Then for any cẠ Z0, f), Hie] we define the H-valued linear map

(4.4) u,(2) =| PrB,E(z) 0 dz, A A dn Ở

-{ xÈỂ) A đại A A đzm

4

Let us remark that the map (4.4) may be not null only on the space Z"{(6, 0),H*] of those forms of Z|[(ụ, 0), H/9ồ] which are homogeneous of degree min Gy, ~ +5 GqỪ br) ề+ +Ừ CmỪ Indeed, if Ạ is of degree < m Ở1 then, by Corollary 3.4

B,é is of degree < mỞ2 and the integral (4.4) is plainly null When Ạ is of degree >m-+ 1 then P7é = 0 and the part 4,1 of B,ằ of degree mỞ1in Ất, .; 6m; which is theỢỖonly one participating at the integration, is of degree >1 in o,, .,0,,

hence Pry,,-1 = 0

4.1 LEMMA The map pu, does not depend on the particular choice of the set A> &,(a, H) and is continuous

Proof Indeed, if yn ằ ZỢ ((6, 0), H, loc] and its support is compact and disjoint of f(a, H) then, by Corollary 3.4, B,y satisfies (6,+ 0)B.n = nin Uand the support of B,y is contained in the support of 4 Then wecan write Py = On, in U, where the support of 7, is compact, hence by (4.3)

\ Prn(z) A dzy A A dz_=0

U

In particular, if we take

n = ((đ, + ửyB,Ọ Ở È) Ở ((, + 9)w;B,ẤỌ Ở ẹ),

where J; e C*(U) is zero in an open neighbourhood of F (a, H) and is one outside

another (relatively compact) open neighbourhood of /,(a, 7) in U(j= 1, 2), then

we obtain that the integral (4.4) does not depend on 4

A similar argument shows that we have also, for any ẠẠ Z"[(c, ằ), Hi],

LCE) =| PR, o6(Z) A dz A A dz, Ở

Ở| Pré(z) A dz, A A dZy;

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where Q > A is a relatively compact open set in U, whose closure is disjoint of SF (4, H), and R, gq is given by Theorem 3.7 If y Ạ Cậ*(Q) has the property that y=1 in a neighbourhood of Z then by (4.2)

\ PER, of(z) A đối A .A đzy =( P20 WR, of(z) A dz, A A dz

= 1

Since R, 9 is continuous and OR, 9 = Ạ Ở 5,R,,9ằ in Q, we infer the continuity of

uẤ on ZỢI(ụ, 0), Hie],

Let us mention that the continuity of Ấ can be also proved by showing that B, is closed, hence continuous, on Z[(o, ẹ), Hi] However, the formula (4.5) makes a connection between yu, and Theorem 3.7

Note that Z[o, Ô, Hự*] is an A(U)-module In some important cases the

space H can be also given a structure of A(U)-module by means of the maps (4.6) v,( fox = u,{fxo, A AG,) (ẶcA(U); xeH)

The map yp, itself becomes an A(U)-module homomorphism More precisely, let us

denote by Z@[(c, ặ), Hi] the closure in ZỖ[(a, Ọ), HỮ*] of those exterior forms ầ} 7;x;,

J

where the coefficients of n; are smooth functions having vaÌues 1n the commutant

of the set {Ủ¡Ể), , %(z); ze U} in ⁄(H), x;e H and (6, + O)y; = 0 for any j

Then we have the following

4.2 THEOREM Assume that a=(q, ,@,) is a commuting system of operators in Z(H) and define (2) = z¡ Ở d, j= Ì, ,H, Z Ở (Zy, , zẤ) CC" and a = (%, ,%,) If Uc C" is any open set containing Pca, H) then for any

Ee Zél(o, 0), Hie] and fe A(U) we have pf) = vol fig):

It is known that (4.6) provides, in this case, an analytic functional calculus, 1Ạ vy, iS a continuous homomorphism of the algebra A(U) into the algebra F(A) such that v,(p) = p(a), for any complex polynomial p [6}, since Wcn(a, H) is exactly the joint spectrum of the system a = (a,, ., @,) when acting on H [4], [6] Theorem 4.2 asserts that the usual multiplicativity of the analytic functional calculus is, in fact, a property of A(U)-module homomorphism This feature will follow from a Fubini type property of the integral map (4.4), which will be described in the sequel (see also [4], [6])

Suppose that U c C"*Ỏ is open and write a point in U as a pair (z,w) = = (2, ~~ +5 Za: Way ++ +> Wm) The corresponding system of indeterminates when

defining 0 will be (ặ, 6) = (f,, ., f,, @, -, ẹ) We shall consider two systems

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202 F.-H VASILESCU

A closed set F < U is said to be C"-compact in U if for every compact K c V the set FN (C" K) is compact, where V is the projection of U on the last m coor- dinates [4]

Let us extend the definition (4.4) for commuting systems a = (a, .,%,) with Zu(, H) C"-compact in U Namely, we shall fix an arbitrary function ge CỎ(U) such that p=0 in a neighbourhood of 7,(ề, H), @ = 1 outside another neighbour- hood of /,(x, H) and supp (1 Ở @) is CỖ-compact in U Then the support of the

form (5, + 0)oB,é Ở é is CỢ-compact in U and if P, is the map of Z"[(o, f, 0), Hie

which annihilates the monomials containing g,, ., ằ, and letting the other invariant (which agrees with Pz from (4.4)), then we define

(4.7) tu(Ế) (w) = ( P(O@B,e(z, w) ỞE(z, w)) Adz, A Adz,

cn

for any ẠẠ ZỖ[(o, Ọ, ử), HịP*] We shall see that the definition (4.7) is independent

of g and w Ở u,(Ạ) (w) is analytic

4.3 Lemma Assume that ne A[(ặ, &), Hig] is in the domain of 0 and its

support is C"-compact Then the form Wc w)Adz,A Adz, is in the domain

ụẶ ỏ in A[@, HP] and

nn

2 ẲnG, Wa der Adz, =Â ÔNG, w)Adz,A .Adz

where V is the projection of U on the last m coordinates

In particular, if the degree of ninẠ,, ,Ạ, is <n Ở1 then \ ane, w)A dz, A A Adz, = 0

Proof Take z, Ạ Cằồ(C"*Ỏ) with the properties of Lemma 3.5 Then for every relatively compact open set Qc V we have 4,=y,* 7-27 and On, > On in

A[(E, ẹ), Hy] as ằ>0, where W= Un(CỖxQ) If n=n,-1-+7,, where n,-1 is of degree <n Ở 1in{,, , ẹ, then we have for we Q

(ine, wyAdzA A dz,= tim (ont, w)A đZ¡A A đZẤ =

e70

lim \ ng, w)Adz,A Adz,=limd Wn, 2,1) den 1 Adz

630 e+0 n?

where Nn-1, 2 = Le * Mn and Na,e = ke * NnỪ since

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for a sufficiently small ằ, by the Stokes formula Consequently, \4(z, w) A dz, A

A Adz,= \ f(z,w)A dzyA A đzẤ is in the domain of @ in Ala, H's

and the first assertion holds The second assertion follows easily from the first 4.4 LEMMA The map yu, given by (4.7) does not depend on the particular choice of the function p Moreover, u,(Ạ) Ạ A(V, H), for every Ee Z[(a, Ạ, ệ), HỊP"| Proof The independence of yu, on the choice of @ follows by refining the first part of the proof of Lemma 4.1, via Corollary 3.4 and Lemma 4.3 Note also that the

integrand of (4.7) satisfies

OP,(0pB,é(z, w) Ở E(z, w)) = Ở P,0E(z, w) = P,6,ằ(z, w) = 0,

hence, by Lemma 4.3, u,(é) = 0

The next result is an extension, in Hilbert spaces, of Theorem 3.6 from [4] 4.4 THEOREM Let U be an open set in C"+Ỏ and V the projection of U on the last m coordinates Consider also a = (a, .,%,)< A(U, L(A)) and B = (B,, ., B,) cÁ(F, (H)) such that (a, B) is a commuting system If Sa, H) is C*-compact in U and S,(B, H) is compact then for every Ở Ạ Z"[(a, Ạ, &), Hie] and n Ạ AỢ[(t, 6), CỢ(V, (a, B))], with (6, + 0)n = 0 in V, where (2, B is the commulant of (%, B) in L(A), we have

Hea, ^ é) = HạữnH,(Ế))

Proof The formal part of the proof is not essentially different from that of Theorem 3.8 from [6], which in turn is an explicite variant of the proof of Theorem 3.6 from [4], so that we only sketch it

Consider g ằ C*(U) with the properties from the definition (4.7) for Wy(a, A) Analogously, take we Cồ(V) for ⁄y(Ữ, H) and 0e Cệ(U) for #,((a, 8), H), with similar properties Then we can obtain the relation

DOP 6, ỪBoa, (NA) Ở Pann Pf =

OY PBs (nA (09P,B.g Ở Pot) Ở Pind (O9P,ByE Ở Pof) + Ông,

where the support of m9 is compact By integrating this relation and using the equality

av) Ban A (OpP,B,Ạ Ở Pi)Adz,A Ad02, =

=0 B,n \(BụP,BẤỌ Ở P,é)adza Adz,+ Om,

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204 F.-H VASILESCU 4.5 Proposition Let Uc C"*Ỏ be an open set and x = (%, ,%,) < ACU, ặ(A)) is a matrix of commuting elements which commute also with 21, , 3Ấ

Denote by ¡ = 3 uy, = (fu , Ba) JẶ ⁄v(x H) and v(, H) are both k=1

C"-compact then u.(2) = p,(uc), for any Ee Z"[(o, Ạ, ệ), Hi], where ự ¡is the map

induced by the formula

U(z,w)(xo,A Ao,)= 3; (det uy, 5,(2, Wn, gX Ou, A + A Ox,

peer <kp

for xẠ H, u(z, w) being the identity on the other terms (here ỘỔdetỖỖ stands for the determinant)

The proof of Proposition 4.5 is similar to that of Proposition 3.12 from [4] (see also [6]), so that we omit it

Proof of Theorem 4.2 By Lemma 4.1, it will be sufficient to verify the property uẤ(f6) = v,(Ặ) nẤ(Ọ) for Ọ = nx, where the coefficients of 7 are smooth functions with

values in the commutant of ề in #(H) and xe H By Theorem 4.4 and Lemma 4.1 we can write

val f) Ba(nx) = AA) nx) = By(n Val fx) = How, aN AẶXử; Nuwar A 6,) If we transform the system (z, Ở đị, , ZẤ Ở Gy, W; Ở @1, -; W, Ở a,) into the system (wy Ở Zạ, ;, Mạ Ở Zm Wi Ở Qy, -5W, Ở4,) with a suitable matrix (see Theorem 4.3 from [4] or Theorem 4.1 from [6]) then the form 9A fxo,A

Aa, remains unchanged and we obtain by Proposition 4.5,

Ha, a(M A FXO, A A On) = Man ve(f) Xx) = He fnx),

where Ặ{z, w) = w Ở z and /(Ặ) (w) = Ặ(w) J6]

Theorem 4.2 is, of course, related to the existence of the analytic functional calculus for commuting systems in #(H) Moreover, the formula (4.4) is given by a canonical kernel of Martinelli type (see also [6]), while (4.5) is connected to a family of canonical kernels, seemingly depending upon the parameter 2 However, Theorem 4.2 gives more than the analytic functional calculus, actually in the one-dimensional case Let us illustrate this assertion

Consider b Ạ f(H) and take an open set U < C containing the spectrum of b In this case the space Z'[(a, 2), H{?] can be identified with the space of those pairs

(fy fs) from Hie* x Hie* such that (0f,/82Z)(z) =(z Ở b) f2(z) almost everywhere in U If we denote by & the form fio + fot then for a(z) = z Ở b the formula (4.4) can

be written as

p(é) = ( (ằ Ở by fz) dz + ( fal) dzn dz,

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where 4 c U contains the spectrum of 6 and I is its boundary By Theorem 4.2 we

obtain p,( fé) = f(b)u,(ằ), for any fe A(U) and fe Z3[(o, 0), Hie], where f(b) is

given by the Riesz-Dunford functional calculus In fact, this property is valid in this case for a larger class of forms, namely for those obtained by asking only the integrability of ft

In the scalar case, the formula (4.4), more precisely an extension of it, is connect- ed to an integral representation formula of Martinelli type with an additional term It is therefore plausible that these techniques can be applied in order to obtain inte- gral representation formulas of a more general type, in particular for exterior forms whose coefficients are vector-valued functions, with respect to commuting systems of operator-valued analytic functions

REFERENCES

1 FOLLAND, G B.; Kounn, J J., The Neumann problem for the Cauchy-Riemann complex, Prin-

ceton Univ Press and Univ of Tokyo Press, Princeton, New Jersey, 1972

2 HORMANDER, L., An introduction to complex analysis in several variables, D Van Nostrand Co., Princeton, New Jersey, 1966 ồ

3 Koun, J J., Harmonic integrals on strongly pseudoconvex manifolds, I, 0, Ann of Math.,

78 (1963), 112Ở148; 79 (1964), 450Ở472

4 TAyLoR, J L., The analytic functional calculus for several commuting operators, Acta Math.,

125 (1970), 1Ở38

5 Treves, F., Topological vector spaces, distributions and kernels, Academic Press, New York, 1967

6 Vasicescu, F.-H., A Martinelli type formula for the analytic functional calculus, Rev Rou-

manie Math Pures Appl., 23: 10 (1978), 1587Ở1605

7 VASILESCU, F.-H., Special perturbations of the ậ-operator, Communication presented at the First Romanian-American Seminar in Operator Theory, Iasi, 1978

8 WARNER, F W., Foundations of differentiable manifolds and Lie groups, Scott, Foresman and Co., Glenview, London, 1971

F.-H VASILESCU,

The National Institute for Scientific and Technical Creation,

Department of Mathematics, Bd Pacii 220, 77538 Bucharest, Romania