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AN OPERATOR BOUND RELATED TO REGULAR OPERATORS Brian Jefferies and Susumu Okada Abstract. Regular operators on Lp -spaces are characterised in terms of an operator

bound which is associated with certain generalisations of the Feynman-Kac formula.

Let (; S ; ) be a measure space and let T : L2(; S ; ; C ) ! L2(; S ; ; C ) be a bounded linear operator. Let Q be the spectral measure acting on L2(; S ; ; C ) of multiplication by the characteristic functions of elements of the -algebra S , so R that if f is a bounded S -measurable function, Q(f ) =  f dQ is the operator of multiplication by f . When is it true that we can nd a positive number C such that the inequality (1)

k

X

Q(gj )TQ(fj )

j =1



k

X C

fj

j =1



gj

1

;

holds for all bounded C -valued S -measurable functions fj ; gj ; j = 1; : : :; k; de ned on  and all k = 1; 2; : : :? Here k
k is the operator norm on the space of bounded linear operators acting on L2 (; S ; ; C ), u v denotes the function (u v)(x; y) = u(x)v(y); for all (x; y) 2 R 2 , and the norm k
k1 denotes the L1 -norm with respect to   on   : Another way of stating inequality (1) is that the bilinear map (f; g) 7! Q(g)TQ(f ); f; g 2 L1 (; S ; ; C ) is continuous for the topology of bi-equicontinuous convergence. The operator bound (1) arises in the construction of an operator valued set function Mt from a semigroup S of operators and the spectral measure Q. If bounds of the form (1) hold, then the additive set function Mt is bounded on an underlying algebra of sets, as outlined in [3]. It turns out that ideas arising in the theory of complex vector lattices are relevant to the characterisation of operators T for which the bound (1) holds, namely, regular operators. In [J], an additional assumption was made that the operator T is a Fourier multiplier operator acting on an Lp -space of a locally compact abelian group, in which case more properties than in the present situation can be established. However, operators which arise in applications are often not Fourier multiplier operators. Section one introduces some terminology from vector lattices in the context of p L -spaces. The main result of the present note, Theorem 1, is presented in section 1991 Mathematics Subject Classi cation. Primary 47A30, 43A2; Secondary 81S40, 35L45. Typeset by AMS-TEX 1

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BRIAN JEFFERIES AND SUSUMU OKADA

two, where the characterisation of operators T for which the bound (1) holds is also given for the case of Lp -spaces 1  p  1. For the examples in Section 3, we prove in Proposition 1 that regular operators T which are given locally by an integral kernel K , the kernel of the modulus jT j of T is given by jK j|a slight generalisation of [5, p288]. The examples of Section three illustrate the range of applicability of the characterisation given for operators satisfying (1). The arguments that follow can be modi ed suitably to apply to a continuous linear map T between Lp -spaces over dierent measures spaces, but for ease of presentation, we assume that the measure spaces of the domain and range of T are identical. 1. Complex vector lattices.

Let (; S ; ) be a measure space and let F be either the eld C of complex numbers, or the eld R of real numbers. For 1  p < 1, Lp (; S ; ; F) denotes the vector space of equivalence classes of F -valued S -measurable functions which are pth-integrable with respect to , modulo -null functions. For the case p = 1, the space of equivalence classes of bounded F -valued S -measurable functions, modulo functions vanishing locally -almost everwhere is denoted by L1 (; S ; ; F); a function f vanishes locally -almost everwhere if the set E = fx 2  : f (x) 6= 0g is a local -null set, by which we mean that (A \ E ) = 0 for every set A 2 S with nite -measure. The order relation f  0; f 2 Lp (; S ; ; R); also written as 0  f , is taken to mean that f is the equivalence class of a function greater than or equal to zero -almost everywhere, in the case 1  p < 1, and locally -almost everywhere for p = 1. The spaces Lp (; S ; ; C ); 1  p  1, are complex vector lattices [4, De nition II.11.1] with respect to these order relations. A subset A of Lp (; S ; ; R) or Lp (; S ; ; C ) is said to be order bounded if there exists u  0 such that u  jf j for all f 2 A. If every order bounded subset A of Lp (; S ; ; R) has a supremum, then Lp (; S ; ; R) is said to be Dedekind complete . The spaces Lp (; S ; ; R) are Dedekind complete for any measure space (; S ; ) in the case that 1  p < 1 [4, Proposition II.8.3, Exercise II.23]. The space L1 (; S ; ; R) is Dedekind complete if and only if (; S ; ) is a localisable measure space [4, Exercise II.23], see [4, pp157-8] for the de nition. In particular, Lp (; S ; ; R) is Dedekind complete for every 1  p  1 if (; S ; ) is a - nite measure space. De nition 1. Let 1  p  1. A bounded linear operator T from Lp (; S ; ; F) to Lp (; S ; ; F) is said to be positive if for every f 2 Lp (; S ; ; R) such that f  0, we have Tf  0. A bounded linear operator T : Lp (; S ; ; F) ! Lp (; S ; ; F) is said to be a regular operator if it can be written as a linear combination of positive operators. Let 1  p  1. The restriction TR of a bounded linear operator T : Lp (; S ; ; C ) ! p L (; S ; ; C ) to Lp (; S ; ; R) may be written as TR = T1 + iT2 for bounded real linear operators T1 ; T2 acting on Lp (; S ; ; R). In the case that Lp (; S ; ; R) is a Dedekind complete normed vector lattice, T is regular if and only if both T1 and T2 map order bounded intervals in Lp(; S ; ; R)

AN OPERATOR BOUND

3

into order bounded intervals, in which case the positive linear operators T1 ; T2 on Lp (; S ; ; R) are de ned by setting, for each j = 1; 2,

Tj+ f = supfTj u : 0  u  f g;

Tj? f = supf?Tj u : 0  u  f g;

for every f 2 Lp (; S ; ; R) with f  0. Then TR = T1+ ? T1? + i(T2+ ? T2? ). The modulus jT j : Lp (; S ; ; C ) ! Lp (; S ; ; C ) of a regular operator T is de ned by the formula jT ju = sup0jf ju jTf j; for all u  0 [4, De nition IV.1.6]. 2. An operator norm inequality.

The space of continuous linear operators acting on a Banach space X is written as L(X ). We rst observe that the operator norm inequality (1) sometimes holds automatically. Proposition 1. Let (; S ; ) be a measure space, p = 1 or p = 1, and let T be any bounded linear operator acting on Lp (; S ; ; C ). Then the operator norm bound (1) holds for all bounded complex valued S -measurable functions fj ; gj , j = 1; : : :; k; and k = 1; 2; : : : . Proof. The L(Lp(; S ; ; R))-valued set function

m : A  B 7! Q(B )TQ(A);

A; B 2 S ;

is additive on the semi-algebra of all product sets belonging to S , so it is the restriction to the family P of product sets of a unique additive set function on the algebra a(P ) generated by P . The same symbol m is used to denote this extension. Any set E belonging to a(P ) may be expressed as the nite pairwise disjoint union [kj=1 (Aj  Bj ) of product sets, so that

m(E ) =

k X j =1

m(Aj  Bj ):

Moreover, either the sets B1; : : :; Bk or the sets A1 ; : : :; Ak may be assumed to be pairwise disjoint. In the case that p = 1, assume that B1; : : : ; Bk are pairwise disjoint. Then for each j P = 1; : : :; k, the operator Q(Aj ) maps the unit ball of L1 (; S ; ; C ) into itself and k kj=1 Q(Bj )uj k1  maxj kuj k1 for any uj 2 L1 (; S ; ; C ), j = 1; : : :; k: It follows that km(E )k  kT k for all E 2 a(P ). For the case p = 1, assume that the sets A1; : : :; Ak are pairwise disjoint. Then

km(E )uk1 

k X j =1

kQ(Bj )TQ(Aj )uk1  kT k

k X j =1

kQ(Aj )uk1  kT k kuk1 ;

for all u 2 L1(; S ; ; C ): Consequently, km(E )k  kT k for all E 2 a(P ).

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BRIAN JEFFERIES AND SUSUMU OKADA

For any a(P )-simple function , that is, a nite linearR combination of characteristic functions of sets belonging to a(P ), the integral   dm is de ned by linearity, and according to [2, Proposition I.1.11, Theorem I.1.13], it satis es

Z





 dm

 4kT k kk1 :

Moreover, the same estimate holds if  is the uniform limit of a(P )-simple functions. In particular, if k = 1; 2; : : : and fj ; gj ; j = 1; : : :; k; are bounded C -valued S Pk measurable functions de ned on , then the function  = j=1 fj gj is the R P uniform limit of a(P )-simple functions and   dm = kj=1 Q(gj )TQ(fj ): The bound (1) follows.  In the case p = 1, any bounded linear operator acting on L1(; S ; ; C ) is regular according to [4, Theorem IV.I.5] and the remark after [4, De nition IV.4.2]. For p = 1, any bounded linear operator acting on L1 (; S ; ; C ) is regular provided that (; S ; ) is a localisable measure space [4, Theorem IV.I.5]. These are the extreme cases; for 1 < p < 1, we have the following relationship beween the bound (1) and regular operators. Theorem 1. Let (; S ; ) be a measure space, 1 < p < 1 and let T be a bounded linear operator acting on Lp (; S ; ; C ). Then the operator T is regular if and only if there exists a constant C > 0 such that the operator norm inequality (1) holds for all bounded complex valued S -measurable functions fj ; gj , j = 1; : : :; k; and k = 1; 2; : : : . Proof. We consider rst the case in which T is regular. It suces to show that the inequality (1) is true for positive operators, for then it is true for all regular operators, so we shall suppose instead that T is positive. Let u  0 be an element of Lp (; S ; ; R). The Lp (; S ; ; R)-valued set function

mu : A  B 7! Q(B )TQ(A)u;

A; B 2 S ;

is additive. Its extension to the algebra a(P ) generated by product sets is denoted by the same symbol. Any set E belonging to a(P ) may be expressed as the nite pairwise disjoint union [kj=1 Aj  Bj of product sets, so that

mu (E ) =

k X j =1

?




mu (Aj  Bj ) = mu (  ) ? mu (  ) n E :

It follows that 0  mu (E )  mu (  ) = Tu. Thus, supE2a(P ) kmu (E )kp  kTukp , that is, mu is a bounded additive set function on the algebra a(P ) of subsets of   . As in the proof of Proposition 1, for any a(P )-simple function , we have the estimate

Z

 4kTukp kk1 ;

 dm u



p

AN OPERATOR BOUND

5

and the same estimate holds if  is the uniform limit of a(P )-simple functions. In particular, if k = 1; 2; : : : , and fj ; gj ; j = 1; : : :; k; are bounded C -valued S Pk measurable functions de ned on , then the function  = j=1 fj gj is the R P uniform limit of a(P )-simple functions and   dmu = kj=1 Q(gj )TQ(fj )u: By decomposing an arbitrary function u 2 Lp (; S ; ; C ) into its real and imaginary parts, and then into positive and negative parts, it follows that the bound (1) holds with C = 16kT k in the case where T is positive. The case of regular T , for which C = 64k jT j k, follows immediately. Now suppose that the bound (1) holds. The spaces Lp (; S ; ; C ) and Lq (; S ; ; C ), 1=p + 1=q = 1 are dual to each other with respect to the duality

hf; gi =

Z



f 2 Lp (; S ; ; C ); g 2 Lq (; S ; ; C ):

f (x)g(x) d;

We shall show that T is regular and the modulus jT j of T is given by (2)

8 k 0, and so the equality F (u; v1 +v2 ) = F (u; v1) + F (u; v2) holds. The equality F (u; cv) = cF (u; v) clearly holds for all numbers c  0. P

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BRIAN JEFFERIES AND SUSUMU OKADA

Every function belonging to Lq (; S ; ; C ) decomposes into real and imaginary parts, and these components decompose essentially uniquely into positive and negative parts, so there exists a unique linear extension of F (u;
) to all of Lq (; S ; ; C ) such that the bound F (u; v)  4C kukp kvkq holds. Because 1 < p < 1, the space Lp (; S ; ; C ) is the continuous dual of Lq (; S ; ; C ), so there exists an element u of Lp (; S ; ; C ) such that hu ; vi = F (u; v) for all v 2 Lq (; S ; ; C ). By virtue of the inequality hu ; vi  0, true for all v  0 in Lq (; S ; ; R), we have u  0. For every v  0 in Lq (; S ; ; R); and every bounded S -measurable function f such that jf j  1, it follows from formula (2) that jhT (fu); vij  hu ; vi, hence jT (fu)j  u . If T1 and T2 denote the real and imaginary parts of T , it follows that T1 and T2 map order bounded intervals in Lp (; S ; ; R) into order bounded intervals. Under the assumptions, the space Lp (; S ; ; R) is Dedekind complete, so T is regular. From the de nition of the modulus jT j of T , we have jT ju  u . Now suppose that g 2 Lp (; S ; ; R) satis es jT (fu)j  g, for each complex valued S -measurable function 0  jf j  1. Then k X j =1

jhQ(gj )TQ(fj )u; vij 

k X j =1

hQ(gj )jTQ(fj )uj; vi 

k X j =1

hQ(gj )g; vi = hg; vi;

for all v 2 L2(; S ; ; R) with v  0 -a.e., all k = 1; 2; : : :; and all bounded complex valued S -measurable functions fj and gj  0, j = 1; : : :; k such that 0  jfj j  1 Pk for all j = 1; : : :; k and j=1 gj = 1. Then hu ; vi = F (u; v)  hg; vi. Hence, u  g and the equality u = jT ju follows.  Corollary 1. Let (; S ; ) be a measure space, 1  p  1, and let T be a regular linear operator acting on Lp (; S ; ; C ). Then for all u  0 in Lp (; S ; ; R), and all v  0 in Lq (; S ; ; R), with 1=p + 1=q = 1; we have 8 k 0 such that

Z



(4)



f (y) d(y)

jK (
; y)j

p

 C kf kp ;

for all f 2 Lp (; S ; ; C ). If p = 1 or p = 1, then T is necessarily regular. If 1  p  1 and T is regular, then T is a kernel operator, the bound (4) holds and Z jT jf = jK (
; y)jf (y) d(y): 

Proof. By the exhaustion principle [5, Theorem 86.1, p142], there exists an increasing sequence of sets n 2 S with nite -measure such that  = [1 n=1 n and for p each n = 1; 2; : : : , the equality (3) holds for all f 2 L (; S ; ; C ) vanishing outside n . Suppose that 1  p < 1 and the inequality (4) holds. Then T coincides on a dense set with the integral operator de ned by the kernel K . The inequality (4) implies that the latter is regular. For each A 2 S , let Q(A) : Lp (; S ; ; C ) ! Lp (; S ; ; C ) be the operator of multiplication by the characteristic function A of A. If 1  p  1 and T is regular, then TQ(A) is regular for every set A 2 S . Furthermore, for each n = 1; 2; : : : , TQ(n ) is a regular kernel operator, so by [5, p228], the kernel of the modulus jTQ(n)j of TQ(n) is (x; y) 7! jK (x; y)jn (y). Moreover, for all f  0,

jT jQ(n )f = jTQ(n)jf = R

Z

n

jK (
; y)jf (y) d(y):

and n jK (
; y)jf (y) d(y) = jT jQ(n)f  jT jf for all n = 1; 2; : : : . There exists a set 0 of full -measure such that (jT jf )(x) < 1 and Z

n

jK (x; y)jf (y) d(y)  (jT jf )(x)

for all x 2R 0 and n = 1; 2; : : : . By monotone convergence,R for each x 2 0 , the integrals n jK (x; y)jf (y) d(y) converge as n ! 1 to  jK (x; y)jf (y) d(y). But jT jRQ(n)f ! jT jf in Lp(; S ; ; C ) as n ! 1, so (4) holds and we have jT jf =  jK (
; y)jf (y) d(y). 

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BRIAN JEFFERIES AND SUSUMU OKADA

3. Examples The selfadjoint operator D = 1i dxd acting in L2 (R) has the property that eiDt ; t 2 ?
2 iDt

is the group of translations de ned for each f 2 L (R ) by e f (x) = f (x + t) for almost all x 2 R and all t 2 R . For each t 2 R , the operator eiDt 2 L(L2(R )) is regular, although it is not a kernel operator. We know from Theorem 1 that (1) holds. The bound (1) can be computed directly for T = eiDt , because for ; 2 L2(R ), the Cauchy-Schwarz inequality tells us that

R

j

k X j =1

hQ(gj )eiDt Q(fj );

ij 

Z

k

j

k X

R j =1

k X j =1

gj (x)fj (x + t)j j(x + t) (x)j dx

fj gj k1 kk2 k k2;

for all bounded C -valued measurable functions fj ; gj ; j = 1; : : :; k; de ned on R and all k = 1; 2; : : : . The operator eiDt is a Fourier multiplier operator for all t 2 R , so Theorem 1 of [3] applies as well, for eiDt is the operator of convolution with respect to the unit point mass ?t concentrated at ?t. In the following example, we show how Theorem 1 can be applied to a regular operator which is not a Fourier multiplier operator, in which case [3, Theorem 1] is inapplicable. Example 1. Let V : R ! R be a locally square integrable function. Let D(H ) be the set of all elements f of L2(R ) which are absolutely continuous on bounded intervals and Hf := Df + V f belongs to L2(R ). By this we mean that Df + V f is the element of L2 (R) corresponding to the function x 7! 1i f 0(x)+ V (x)f (x) de ned for almost all x 2 R . Then H is selfadjoint and iH generates the unitary group eiHt given for each  2 L2 (R) and t > 0 by the formula Rt
eiHt  (x) = ei 0 V (x+s) ds (x + t);

?

for almost x 2 R , as may be seen by dierentiation. The operator eiHt does not commute with translations if V is nonconstant on a set of positive measure and t 6= 0|under these conditions, eiHt is not a Fourier multiplier operator. Clearly, eiHt maps order bounded sets into order bounded sets in L2(R ), so the bound (1) necessarily holds. The bound can also be computed directly, as above. In the following example, we show how Theorem 1 can be applied to an operator which is not a Fourier multiplier operator, but whose integral kernel is known explicitly. The operator is not regular and the bound (1) fails. Example 2. Let
be the selfadjoint operator acting in L2 (R ) associated with d2 =dx2. Let V (x) = 21 x2 for all x 2 R and H = ? 12
+ V . According to Mehler's formula [1, Theorem 7.13], for all  2 L2 (R) and t 6= n, n 2 Z, eiHt  is given by

AN OPERATOR BOUND

9

R eiHt  = limn!1 ?nn Kt(
; y)(x) dx with ! 2 ? y 2 ? it ?eit x ? y
2 x Kt(x; y) = ?1=2(1 ? e2it )?1=2 exp ? ? 1 ? e2it ; 2

where the limit is understood in the sense of mean square convergence, that is, eiHt is a local kernel operator on L2(R ) with kernel Kt. If eiHt were a regular operator, then according to Proposition 2, the modulus jeiHt j of eiHt would be a kernel operator with kernel jKtj. To simplify matters, suppose that t = =2. Then, !

2 ? y 2 + (ix ? y )2 x ? 1 = 2 = (2)?1=2 jK=2(x; y)j = (2) exp < ? 2

is not the kernel of a bounded linear operator on L2(R ). By Theorem 1, the bound (1) fails to hold for the operator T = eiH=2. 1. 2. 3. 4. 5.

References E.B. Davies, One-Parameter Semigroups, Academic Press, London/New York, 1980. J. Diestel and J.J. Uhl Jr., Vector Measures, Math. Surveys No. 15, Amer. Math. Soc., Providence, 1977. B. Jeeries, An operator bound related to Feynman-Kac formulae, Proc. Amer. Math. Soc. 122 (1994), 1191{1202. H.H. Schaefer, Banach Lattices and Positive Operators, Grund. Math. Wiss., vol. 215, SpringerVerlag, Berlin/Heidelberg/New York, 1974. Zaanen, Riesz Spaces Vol. II, North Holland, Amsterdam/New York/Oxford, 1983. School of Mathematics, UNSW, NSW 2052 AUSTRALIA

E-mail address : B.Je[email protected]

Department of Mathematics, University of Tasmania, Hobart, AUSTRALIA

E-mail address : [email protected]