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ANALYTIC PERTURBATIONS OF THE 0-OPERATOR AND INTEGRAL REPRESENTATION FORMULAS IN HILBERT SPACES F.-H.. INTRODUCTION In this paper we present the construction of some operator-valued ke

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 ¢(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 € @(A) with A(T) œ 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

Ø, NA su AG; J (<j< <<m;p=l, ,n)

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

Let us define the operators

188 F.-H VASILESCU

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

where ¢j -+ a; Éjˆ is the canonical decomposition of an arbitrary element € € A[o], with ¢; and ¢j’ not containing ¢; Note the anticommutations relations

S;S, + S,S; = 0

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 ® 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 Ø

Any commuting system of endomorphisms ø = (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 (€€ 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 €(H) will be extended by T @ 1, denoted simply by 7, defined

on ZT) ® 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 C” and C®(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, £(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 = C™(Q, H) will be then given by

(1.5) ỗ„È() = ((Z)Š +- + %(Z)S,)šŒ), (zeÓ),

where € € A[a, C°(Q, H)] We can consider also the usual Ø-operator

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

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

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

(G,, +,6,, 2,, ,dz,,), and has the property (6, + 0)? = 0, since 5,0 =—06,

The aim of this paper is to study the exactness of the operators of the type

6, +0, as well as some of its consequences, in certain Hilbert spaces of square

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 ¢(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 C” 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 £7(2Q) @ H of the tensor product L*(Q) ® 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 L°(Q)

Let us fix a system of indeterminates € = (¢,, ., ¢,,) 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 & € A[Z, Hg] can be associated with a A[Z, H]-valued distri- bution v; by the formula

(2.1) vp) = \ ọ(z)£) d2(z), (p C#(®)),

where dÀÄ ¡is the Lebesgue measure and C?°(Ø) is the usual subspace of C™(Q) (= C°(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

due =(—“-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 ¢ € A[Z, Hy] such that there

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

0.2) (ø2)2)4/)= —( x (i+ + = 3) ^ ÉŒ) đÃ(z)›

190 F.-H VASILESCU

for any g € 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,, ,¢,,) 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 € is in A[Z, C°(Q, H)] and both ¢ and dé belong to Ate, Hl, where 0é is defined

as in (1.6) with an obvious identification, then € € G(@) and € satisfies also (2.2)

A nother useful remark is that if » ¢ A[f, C°(Q, Y(H))] and y with its derivatives are bounded on @ then for any € € BA) we have also pa €G(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 @ € 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 Ø(2) c # (0), as one can see from the formula (2.2) Therefore 0* has a similar property Let us denote

by 0, the “scalar” 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 €€ (0) there is a sequence ẽ,e 2(ô,) @ H such that

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

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

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

Proof Let us fix € € 9A) Since the coefficients of € 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 € = 3 ÿ;„, where J = (i,, , 7,)

Lf

is an arbitrary multi-index with | <i, < <i, <m and fp= Ci, A AC (The symbol ‘‘@’’ 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) = » <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) = \ ø(2) Ð (n2), eÈ su, đA(z) =

I

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

f

for any ø e C§°(@) In this way u„š e 2() and đuyế = u,0€.

J—ẲŒœ

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

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

inequalities Therefore the Lebesgue theorem of dominated convergence implies that š, € and n, = 0¢, > n = 0¢ as j — 00, in Alt, Ho)

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

7 define the operator

wi ¥2707) = Yc e,Sš; € A[E, C]

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

i

\ 9) ¥; <n), ed) dA) =

i

=wy y § leone) ax) 0 = I

= —» ( x Đế (Ez) daz) Er ) =

= —[do@) 0 Yi <E@), 0) CG),

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

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

of Lemma 2.1

Consider now é€Q(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 ¢; = ¥, 4 is

kel

an element of 2(Ø*) since

<ế;, 0y) = (y uy! tế, rỒ,

k=1 for any y € (A), on account of the first part of the proof The same argument using the Lebesgue theorem of dominated convergence shows that ¢; + ¢ and 0%, = Oe

as j > oo, in All, Hol)

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

J easily that 9@x eA) for any xe H, therefore assuming € = VY ¢,f, and OE =

+ i

192 F.-H VASILESCU

= }) n,¢, we shall have

€3 ›<66), e2 bu 9, 38,0} =

=©; E1(#), &> e675 98,80) =

= <nil*), > ey Ấn, Y0 @exbs> =

=h ny(*), n> Cy, 3,6;

hence ¥} <Z,(*), Sš, 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 Ø 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,£)@x is in GO*)@H then for any (¥ é,f,)@y in

BO) @H we have

‹(*0;)@x, ay Ef)@y =

‹(ð* 3.0,,)@*x, (40)

7 and approximating any Š e 2(2) with elements from B(0,)® 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 C” (in the sense of [l]) then the range

of 0 in A[£, 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) ¢ #(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,

herefore (Z + 1)7 is self-adjoint, whence L is self-adjoint

2.4 LemMMA The operator B = T + T®* is self-adjoint

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

Proof The equality (2.4) follows from the relation

whose proof is straightforward

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

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

latter by Lemma 2.3) we have also that L @ 1> 00* + 0*0, hence they must coincide In particular, the range of 00* + 0*@ is closed in Alp, Hg] Since 0 + ử*

is self-adjoint (Lemma 2.4) and the range of a self-adjoint operator is closed if and only

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[€, 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 C” 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,, ,¢,) The system

Ễ = (Ế¡, -; 6„) will be associated, in the sense of the previous section, with the operator 0 It is known that for z¢ 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 C°(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, ©), 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+¢ Y(H) then, by Lemma 2.6, @(T) is closed and H= R(T) © 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, (xeØ(T)),

(ŒT+T*1T*y= TỰ + Ty, (ye WT)

Proof Wf ve A(T) then v= Tv, wÍth pụcÝ(T*), hence (7+ T*) 1ð =

= v, € #(T*) This means that (T + T*)14(T) < (T*) Analogously, we have

(T+ 7T*)'#(T*) c #ŒT).

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

(T + T*)?°Tx = (T ~ T*)1Tx, = (T+ T*)(T + T*)x, =

= (T + T*\T + T*) x, = T*(7 + T*) '!xị = T®ŒT — T®) 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 < C”™ 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[(ø, ©), 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 €D(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 ïy + + 4,, where n, is of degree j in G, , Cm and of degree p — jin o,, .,6,; moreover, by (2.2), each y, is in (0) We shall be looking for a solution € of the equation D,é = yn, where € = 9 + ¢, + + Spa

€; being of degree j in f,,.-.,¢, and of degree p —j — 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 — Ì),

with the conditions

Š „1a =0

ô„n; Ôn,-ạ —= 0 (j=1, ,p)

ôn, = 0

Let us define QO(a(z)) = (6,.) + Ox.) 4 (z € 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 ¢,(z)¢.#(6x.)), for every

z€Q Moreover, as ny € ZA) we have also €,< D(A) and 506) = —05,£5 = —Ony =

= ôm, whence 6,(n, ~0é)) = 0 Define then £&,(z) = O(a(z))(n,(z) — 0&)(z)) €

EH (Oxo) (Z€ Q), hence 6.6, =m —0€ We have also 5,£, € G6), therefore 5,(Y — 0€,) = Syn, + 08,, = 0, which allows the continuation of the procedure

One has, in general, ï(z) — 0Š; ;(2)e2Ý(6„;), hence E(z) = OCa(z))\(nz) —

— 8£ ;—2)) e⁄(š⁄„) and 6,€ + OE 4 =; for any j=1, ,p—1 Note also that š;_¡ c2(6) implies that é,¢ 90) too From the structure of the operator 6, and from the Corollary 2.2 we obtain that the degree of €; in o,, ,0, must be p—j—1 In particular, the degree of y,(z) — 0&,-,(z) in 0, .,0, is zero Since 5,4(-) is exact, the kernel of 6, on the space A°[o, A[C, HJ] must be zero, therefore

OE, 4 =p, and the proof is complete

196 F.-H VASILESCU

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

property Dy =0 in Q then

Bae) = % Š —1*0GG)(Ø0œG)ÿ*n,s@), œ9) 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%,, -,€,, Moreover, By € A?“ {(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

acting in A[Š, C®(Ó, H)], where @ c C” 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 C™ is open and relatively compact I,

ny € All, Ho) has compact support and n € 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 € C&(C”) such that supp x = {z; ||zl| < 1} (where “‘supp”’ stands for the support), y(z) = x(—z), 7 2 0 and \ y(z) dA(z) = 1 Define then y,(z) =

=e ?my(z/e), for any ¢ > 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 € CS°(Q) is arbitrary then a direct calculation from the definition of 0*

shows that

Z#⁄+ -+ se Ban + worn

@.2) 2*(jnq)= — ( 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 — 0

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

write

<x), OY = <p, 448) = <n, y,*O)> =

= (n, 7,*00> = <n, 00> = (*n,, 8,