dirichlet operators on certain banach spaces kato's …math.fau.edu/long/long1998saa.pdfcal...

This article was downloaded by: [Florida Atlantic University]On: 18 May 2015, At: 12:17Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Stochastic Analysis and ApplicationsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lsaa20

Kato's inequality and essential self-adjointness ofdirichlet operators on certain banach spacesHongwei Long aa University of Warwick , Mathematics Institute , Coventry, CV4 7ALPublished online: 03 Apr 2007.

To cite this article: Hongwei Long (1998) Kato's inequality and essential self-adjointness of dirichlet operators on certainbanach spaces, Stochastic Analysis and Applications, 16:6, 1019-1047, DOI: 10.1080/07362999808809578

To link to this article: http://dx.doi.org/10.1080/07362999808809578

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in thepublications on our platform. However, Taylor & Francis, our agents, and our licensors make no representationsor warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Anyopinions and views expressed in this publication are the opinions and views of the authors, and are not theviews of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should beindependently verified with primary sources of information. Taylor and Francis shall not be liable for any losses,actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoevercaused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/loi/lsaa20

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/07362999808809578

http://dx.doi.org/10.1080/07362999808809578

http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/page/terms-and-conditions

STOCHASTIC ANALYSIS AND APPLICATIONS, 16(6), 101 9-1 047 (1 998)

KATO'S INEQUALITY AND ESSENTIAL SELF-ADJOINTNESS OF DIRICHLET OPERATORS ON

CERTAIN BANACH SPACES

Hongwei Long

Mathematics Institute, University of Warwick Coventry CV4 7AL

Abstract

In this paper, we study Dirichlet operators on certain smooth Ba- nach spaces. We establish the well-known Kato's inequality in our general infinite dimensional setting. By applying this,we show the essential self-adjointness of Dirichlet operators with non-constant diffusion part on certain smooth Banach spaces. We also provide an approximation criterion for essential self-adjointness of Dirichlet operators with identity diffusion part on M-type 2 Banach spaces via the classical notion of semi inner-product.

1 INTRODUCTION The essential self-adjointness of Dirichlet operators on finite dimensional or infinite dimensional spaces is a classical and important problem in classical analysis and stochastic analysis. For historical comments and related references we refer to the monograph (131 by Berezanskii. The essential self-adjointnesss of Dirichlet operators on infinite dimensional state spaces has received a lot of attention in recent years. Progress has been made mainly along three lines. The first one is to consider those Dirichlet operators

1019

Copyright O 1998 by Marcel Dekker, Inc.

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5

1020 LONG

with a smooth or measurable drift term which is exactly the vector logarithmic derivative of the Radon measure p on a rigged Hilbert space by using the basic theory of stochastic differential equations and the Cauchy problem of parabolic equations on Hilbert spaces (cf. Albeverio, Kondratiev and Rockner [4-61, Kondratiev [31-321, Kondratiev and Tsycalenko [33] and Liske- vich and Semennov [36]). The second one is to discuss perturbed Ornstein- Uhlenbeck operators on abstract Wiener spaces by using the (r,p)-capacity theory (cf. Shigekawa [41]) and some elementary inequalities for resolvent operators (cf. Song [43]). The third one is to consider Dirichlet operators on path spaces or loop spaces with respect to Wiener measure by establish- ing the hypoellipticity of the Dirichlet operator (cf. Aida [2]) and applying Wielen's technique (cf. Acosta [I]) . In this paper,we would like to extend some known results concerning the essential self-adjointness to a more general class of Dirichlet operators with non-constant diffusion part on certain smooth Banach spaces. The crucial ingredient is the smooth Banach space theory developed in the past two decades. The paper is organized as follows. In section 2, we shall present some basic definitions and results about certain smooth Banach spaces such as the so-called CP%mooth Banach spaces and M-type 2 Banach spaces which were introduced by Wells [45] and Pisier [40] respectively. In section 3, we shall prove the well-known Kato's inequality for Dirichlet operators with non- constant diffusion part in our general infinite dimensional setting. Finally in section 4, we apply Kato's inequality to show the essential self-adjointness of Dirichlet operators . Meanwhile, we also establish an approximation criterion for essential self-adjointness of Dirichlet operators with constant diffusion part on M-type 2 Banach spaces via the classical notion of semi inner- product.

2 SMOOTH BANACH SPACE THEORY

In this section we would like to introduce some basic definitions and results about smooth Banach spaces which will play an important role to deal with Kato's inequality and essential selfadjointness of our Dirichlet operators. Most are abstracted from the recent monograph [25] by Deville, Godefroy and Zizler. Let E be a topological space and A a collection of real valued functions on E.

Definition 2.1. Let {U,) be an open covering of E. W e say that ($1~1 is a A-partition of unity subordinate to {U,) zf the following conditions are satisfied: (1 ) $i E A for each i ; (2) there exists a locally finite open cowering {V,) subordinate to {U,) such that supp$, is contained i n V , for each i;

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5

DIRICHLET OPERATORS ON CERTAIN BANACH SPACES 1021

(3) & ( x ) 2 0 and C$i(x) = 1 for eachx E E. We shall say that E admits A-partition of unity if E is paracompact and for any given open covering {U,) there is a A-partition of unity subordinate to

{Ua>.

Remark 2.2. The above definition is due to Albeverio, Ma and Rikkner [7] which coincides with the classical definition when E is a manifold and A = Ck((k 2: 1) (cf. Lang [35]).

Let E and F be two Banach spaces and Cp(E, F) the space of continuous functions with (k-order (1 5 k 5 p) continuous Fr6chet derivatives. Also denote by CP'q(E,F)(O 5 q 5 p 5 m) those functions in CP(E, F ) whose derivatives of order up to q are bounded. For convenience, we shall use the usual notation C: to replace CPJ'.

Definition 2.3. By a A-bump function on E we mean a nontrivial A- function with nonempty bounded support. E is said to be A-normal if for any disjoint closed subsets K and G of E , there exists a function (o E A such that 0 5 cp < l , (oIK = 0 and (ole = 1.

Definition 2.4. A Banach space E is called CP.4-smooth if there exists a nontrivial CP)q-bump function on E.

Remark 2.5. AYI finite-dimensional spaces and separable Hilbert spaces are Cm3"-smooth. If an LP-space is CP-smooth then it is also Cf-smooth. Note that co is Cm-smooth, but it is not Cz-smooth (cf. Bonic and Frampton [15] and Wells [45]).

Lemma 2.6. If E zs a CP,Q-smooth separable Banach space,then E zs CP7Q- normal. Proof: We prove the lemma by 2 steps. (i)step 1. We first show that E admits CPsq-partition of unity. In fact, we can follow the same argument in Wells [&] to prove this claim and only need to replace BP,q(E, R) and Bp,q(Rk) etc by CP+7(E, R) and Cp,q(Rk) etc respectively. Thus we omit the detailed proof in here. (ii)step 2. Since E is normal, for any disjoint closed subsets K and G of E, we can take an open subset A of E such that K c A c A c E\G. Then {A, E\K) is an open covering of E. Thus according to step 1, there is an Cp!q-partition of unity {fi, f2) which is subordinate to this covering such that suppf, c A and suppf2 C E\K. Then fi is the required normal function.

Definition 2.7. Let E be a Banach space and 1 1 . 1 1 be a norm on it. We say that [I . 11 is an uniformly smooth norm if the limit

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5

lim llx + thll - llxll t-0 t

LONG

exists for each x E SE = { x E E ; llxll = 1) and h E SE and is uniform in ( x , h ) E SE x SE. We say that E is uniformly smooth if its norm is uniformly smooth. The norm 1 1 . 1 1 on E is said to be uniformly convex if lim, llxn - ynll = 0 whenever x,, y, E SE, n = 1,2, . . . are such that lim, llxn + ynll = 2. We say that E is uniformly convex if its norm is uniformly convex. The modulus of convexity of the norm 1 1 . 1 1 is defined for E E [O, 21 b y

6(~) = inf{l- Il(x + y)/211; 2, Y E SE1 112 - yll 1 €1 The modulus of smoothness of the norm ) I . 1 1 is defined fort > 0 b y

We say that E is a p-smooth (1 < p 5 2) Banach space if there exists an equivalent norm on E for which the modulus of smoothness p ( t ) satisfies p(t) 5 CtP for some constant C. We say that E is a q-convex (q 2 2) Ba- nach space if there exist q 2 2 , C > 0 and an equivalent norm 1 1 . 1 1 on E such that 6(&) 1 C E ~ for any E E [O, 21.

Remark 2.8. (1) A norm is uniformly convex on E iff S ( E ) > 0 for all E > 0. (2) 1 1 . / I is uniformly smooth iff limt+o p(t)/t = 0. (3) E is uniformly convex iff E* is uniformly smooth. (4) If E is uniformly convex or smooth, then it is reflexive. ( 5 ) p-smooth Banach spaces are uniformly smooth. For the proof of (2) one can see Lemma 1.3 of Chapter I in [25] by Deville, Godefroy and Zizler. The proofs of (3) and (4) we refer to Proposition 1.11 of Chapter IV in [25]. The proofs of (1) and ( 5 ) are immediate.

Definition 2.9. A Banach space E is called M-type p if there exists a constant c = cp(E) > 0 such that for any E-valued martingale {Mk) the following inequality holds

Remark 2.10. If (E, I I . I ) ) is an M-type p Banach space,then E is apsmooth Banach space. Conversely if E is a psmooth Banach space, then there exists an equivalent norm I / . 1 1 with modulus of smoothness of power p such that (E , 1 ) . 1 ) ) is an M-type p Banach space. These important results are due to Pisier [40].

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5


Theorem 2.11. For a E (Oll]. Assume E admits bump function whose first derivative is a-Holder continuous on E, then E admits a norm with modulus of smoothness of power 1 + a i.e. E is 1 + a-smooth. Proof: See the proof of Theorem 3.2 of Chapter V in [25].

Definition 2.12. For each x E E , we define

J ( ~ ) = (1 E BE*; I(x) = 1 1 3 ; 1 1 ) 1

where BE* = {I E E*; lllllE* 5 1). The multivalued operator J : E --+ E* is called the duality mapping of E.

Remark 2.13. The norm I ( . 1 1 on E is GSteaux differentiable if and only if J (x) is single-valued for any x E E\{O). In this case, the only element of J (x) is the GSteaux derivative ll(x) of 1 at x (cf. Corollary 1.5 of Chapter 1 in [25]).

Proposition 2.14. E v e y CP'q-smooth (2 5 q < p 5 co) Banach space E admits an equivalent norm 1 1 . 1 1 such that (E, / I . 1 1 ) is an M-type 2 Banach space. Proof: Since E is CP,q-smooth ( p 2 q > 2), thus E admits an CP,Q-bump function whose first derivative is globally Lipschitz continuous on E . From Theorem 2.11, we know that E admits a norm 1 1 . ( 1 with modulus of smoothness of power 2 i.e. E is Zsmooth. Then according to Remark 2.10, we conclude that (E l 1 1 . 11) is an M-type 2 Banach space.

3 KATO'S INEQUALITY FOR DIRICHLET OPERATORS WITH NON-CONSTANT DIFFUSION PART ON CERTAIN BA- NACH SPACES

Let (E , 1 1 . 1 1 ) be a separable Banach space and H be a separable Hilbert space which can be continuously and densely embedded into E. Assume that i : H --+ E is the embedding mapping. After identifying E* and H with their images in El we get the following triple

Let [El denote the complexification of a real Banach space e. We can also get the complexification triple [E*] C [HI c [El. Assume that p is a finite Radon measure (e.g.probability measure) on E. As in the previous section, for two Banach spaces E and F we define C;(E, F ) and C&,,(E, F ) as the subsets of CP(E, F ) which are characterized by global or local boundedness of the

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5

1024

derivative up to

LONG

porder in the usual operator norms on E. Also C:(E, F ) denotes the collection of functions in C,P(E, F ) with bounded support. As usual the space of smooth cylindrical functions from E into F is defined by f E 3 C i ( E , F ) if there exist n E N, ( 1 1 , . . . , ln ) C E* and cp, E CL(Rn, F ) such that

f ( ~ ) = c p , ( < l ~ , x > , ~ ~ ~ , < l , , x > ) , x ~ E .

In the space Cz = Cb2 ( E , C ) (C the complex number field), we introduce the Banach norm

Let f be a C-valued function on E. We define the directional derivative of f along the direction h by

ah f ( x ) = lim f ( x + th ) - f ( X I t-0 t

if the limit exists. If there exists a H-valued function D f satisfying the condition

( D f ( x ) , h ) ~ = d h f ( x ) , for all h E H,

we call D f the H-gradient of f on E. Obviously i f f is Frhchet differentiable, then the H-gradient D f is given by D f = i*(df).

Definition 3.1. We say that a measure p is A-differentiable along vector h E E if there exists a measure uh on ( E , B (E) ) such that for all f E A, the following equality holds

If uh << p S O that there is a density Ph of the measure uh with respect to p, then we call it the logarithmic derivative of the measure p along the direction h. Further if p is A-differentiable along all directions h E E* and there exists a Bore1 measurable mapping P : E --, E such that

then this mapping P is called the vector logarithmic derivative of the measure P.

Remark 3.2. The above definition was provided in Norin [39] on locally convex spaces, and was motivated from Fomin [8] and Skorohod [42]. If A is

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5


the collection of indicator functions of Bore1 sets, we obtain diffferentiability in the sense of Fomin. If A is the collection of bounded continuous functions, we obtain differentiability in the sense of Skorohod. In this paper we shall adopt the definition for A = C,'(E, R) ( cf. Definition 7.9 in Bell [12] ). In addition, there is a definition given in Bogachev and Rockner [17] for A = 3C,"(E, R),which was applied to deal with the regularity of invariant measures on infinite dimensional spaces. For detailed discussion and comments on different kinds of differentiability of measures on locally convex spaces we refer to the survey paper Bogachev and Smolyanov [18] and monograph [24] by Daletskii and Fomin. For some extensions to differentiability of a one-parameter family of measures on locally convex spaces one can see Smolyanov and Weizsacker [44]. Many interesting examples can be found in Bogachev and Rockner [17].

~ e t H be an separable Hilbert space. As in Brzezniak and Elworthy [14], we

Put

M(H, E ) := {L : H + E, L is linear bounded and radonifying).

Let UL be the a-additive measure induced by L from the canonical cylindrical Gaussian measure yfi on H. For L E M(H, E ) we put

which, in view of Fernique's Theorem, is a finite number. For a given L E M(H, E) and a bilinear bounded map G : E x E + F , we define

It is easy to prove that

m

T T ~ G = G(LCk, Le-k) := T T ~ L ' G L , k = l

where {Ck)& is an orthonormal basis of H . Let V be an algebra on E which is dense in L2(E, p). Now we define the differential operator H, with domain 2) as follows

1 1 H, f (x) = I ~ ~ g a * ( x ) f" (x)a(x) + - 2 E < b(x), f ' ( 2 ) >E.,

where a : E + M(H, E ) and b : E -+ E are measurable mappings and satisfy the following assumption (C1) 114.)11~(fi,E) E L2(E> P) and Ilb(.)Ils E L2(E> PI. From the argument in section 2 of [3] by Albeverio and Hoegh-Krohn, we

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5

1026 LONG

know that FC," is dense in P ( E , p) for any p 2 l.'Obviously we have the inclusion relation

Fc? c Ct c LP(E, P)(P 2 11, which implies that Ct is dense in P ( E , p) for any p 2 1. In this paper, we can set V to be C;, C i or FC," in terms of different underlying Banach spaces. Further we assume that (H,, Cb2) is symmetric on L2(E, p), i.e.

Note that b should depend on a and measure p in order to ensure that (H,, V) is symmetric on L2(E, p). We shall not discuss the precise conditions on a and b which guarantee the symmetrizability of H, with domain V on L2(E, p). The detailed discussion on this problem we refer to Long 1371. However for convenience we will give some remarks as follows.

Remark 3.3. (1) According to Elworthy and Ma [26] , we have a general abstract result on the symmetrizability of differential operators on L2(E, p) as we describe now. Let u be a strongly V-admissible vector field on E, i.e., there exists divu in L2 (E , p) such that

a:! = -a,f - f d i v v , ~ f E v. Let A be a countable or finite family of strongly V-admissible vector fields on E. Now we define the differential operator H, with domain V by

Hpf (x) = - C a;auf (3) UEA

for all f E V provided that a, f E Dom(d,*). If H, f E L2(E, p) for all f E V, then the differential operator (H,,D) is symmetric on L2(E, p) under certain conditions. From the argument in Introduction of [26] , we know that the above abstract result is also valid for a wide class of separable Banach manifolds endowed with Finsler structure. (2) Let (5, H, E) and (2 , H, E ) be two abstract Wiener spaces (AWS). We assume that A(x) E C ( E ) leaves H invariant for all x in E and satisfies the following assumptions (Al) A E C,1 (E , C(E)) and there exists B(x) E C(H, H ) for each x in E such that the restriction AH(x) of A(x) from E to H satisfies

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5


and AH(x) is symmetric positive definite operator in C(H), i.e. there exists X > 0 such that (AH(x)<, <) 2 X J J < J J L for all x IC E. (A2) There exists a(x) E C(E, E ) for each x E E such that

(A3) For each x E E, the divergence of AH(x) defined by

w

T ~ D A H ( x ) = D A ~ ( x ) ( e k , ek) k = l

exists in H in the sense that the series is absolutely convergent,where {ek)p=o,l is an orthonormal basis of H . We define the second order differential operator H, by

From the fact that for any 6 C(E, E * ) , the restriction of to H is a trace class operator of H ( cf. Theorem 4.6 (Goodman) of Chapter I in Kuo [34]), we know that ~r , ( , ) ; fl'(x) is well-defined. It is well-known that there exists a sequence {P,) of bounded linear operators on E with finite dimensional range contained in E* such that the restriction of P, to H is an orthogonal projection and {P,) converges strongly to the identity operator on H (cf. Lemma 2 in Goodman [27]). In order that the differential operator (H,, C:) is symmetric on L2(El p), we need two additional assumptions on the coefficients of H, (A41 IITTDAH(.)IIH E L2(E,p) and Ila(.)ll&, E L2(E, P). (p l ) II,O(.)IIE E L2(E, p) and ((P,P - ? ( I + 0 as n -t cm in L 2 ( ~ , p),where {P,) is given as the above argument. The assumption (Dl) is satisfied for Gaussian measures and certain measures which are absolutely continuous with respect to Gaussian measures on ( 2 , H, E) ( cf. Proposition 3 in Carmona [19], Theorem 3.5 in Bogachev and Rockner [17] and Corollary 2.1 in Goodman [27]). Under the assumptions (A1)-(A4) and (pl), we can prove that the differential operator (H,, Cz) is symmetric on L2(E, p). In particular, when A = IdE and (H, E) = (H, E), the differential operator H, becomes to be

1 1 Hpf = 1AHf + 5 E < P, f ' >En.

where AH is Gross's Laplacian and /3 : E -+ E is the vector logarithmic derivative of measure p. If ( 2 , H, E) is replaced by a rigged Hilbert space H+ C H c H- with Hilbert-Schmidt embedding mappings, then the differential operator H, reduces to the Dirichlet operator on H+ c H c H- discussed in Kondratiev and Tsycalenko [33] and Albeverio, Kondratiev and Rockner [4-6).

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5

1028 LONG

Lemma 3.4. Let E be a Ct-smooth Banach space, then there exists a sequence of Ct-bump functions { F r ( ~ ) ) r c N such that O < F,(x) 5 1 and

{ 1 for x E {x E E ; llxll I r )

Fr(x) = o for x E { x E E; I I X I I 2 7 + 1)

Proof: Set A, = { x E E ; I1xII 5 T ) , A: = { x E E ; 11x11 < T ) and Br = E\&. Then A, and B,+l are disjoint closed subsets of E. From Lemma 2.6, we know that E is Ct- normal. Therefore Vr E N there exists a function Fr E Ct such that 0 <_ F, 5 1 , FTIA, = 1 and FTIB,+, = 0. The sequence of C:- bump functions {FT(x ) ) is the required one.

In the following lemma, we shall show that Ci is also dense in P ( E , p) on C,4-smooth Banach spaces for p 2 1.

Lemma 3.5. Let E be a C,4-smooth Banach space, then C i is dense in LP(E, p ) for any p > 1. Proof: Since Ct is dense in LP(E, p ) for p 2 1, thus for g E P ( E , p) and any given E > 0 there exists f E C; such that

E L I f - s l p ~ ( d x ) < 5. From the finiteness of measure p on ( E , B (E) ) , we can choose E N large enough such that when r > ro

Now we set f , = F,f, then f , E Ci . It is easy to see that

Therefore from the basic inequality (a + b)P 5 2p-'(aP + bP)(a > 0, b > 0 , p 2 I ) , if r 2 ro, we have

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5

DIRICHLET OPERATORS ON CERTAIN BANACH SPACES

This completes the proof.

Now we state the main result of this section as follows : Theorem 3.6(KatoTs inequality). Assume that E is a separable Ct-smooth Banach space furnished with an equivalent 2-smooth norm I / . 1 1 . Let o and b satisfy (GI) and (C21 a E C&,,(E, M ( H , E ) ) and b E C?.loc(E, E) . I f f : E -. C and H, f are in LP(E, p) (p > I ) (in the sense of the generalized function space (Co2)*), then the following inequality holds

where 0, i f f = O

synf =

We shall prove the above theorem by using the sernigroup method which was introduced in Devinatz [23] in finite dimensional spaces and further developed in Kondratiev [31] and Kondratiev and Tsycalenko [33] in the infinite dimensional setting. Before proving the above Theorem 3.6, we need some lemmas. We follow an approach used in [33] to treat differential operators, which was due to Kato [30] (see also Theorem 1.14 in Cycon, Froese, Kirsh and Simon [ Z O ] ) . First we use the sequence of Ct-bump functions {FT( . ) )rEN constructed in Lemma 3.4 to define a sequence of differential operators with domain Cb2 as follows

Now we can calculate the adjoint of H, by using the syrnmetrizability of (H,, C t ) . Obviously one has

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5

1030 LONG

for f E Ci, when r is large enough, one has suppf C {x E E; llxll 5 T ) .

Therefore

(g*(x)df ( X I , W ) d F , 2 ( x ) ) , -, 0

and

as r --t m. Because C$ 3 F,? + 1, we know that H, f -+ H, f in LP, and so this implies that H: f --+ H,, f in LP ( p > 1). Denote by LO" the space of bounded measurable function on E with the usual supremum norm. In the following crucial lemma, we need to apply the basic theory of stochastic differential equations and Cauchy problems of parabolic equations on M-type 2 Banach spaces. We shall use the related results of Brzezniak and Elworthy [14] and Belopolskaya and Daletskii [lo-111. Even though the results in [lo] were only proved for the so-called T~-class Banach spaces, in fact all the results in [lo] remain true for M-type 2 Banach spaces because the proofs only depended on the well-definedness of stochastic integrals on Banach spaces and this is exactly the case for M-type 2 Banach spaces (cf.[14]). More- over we can replace the Hilbert cylindrical Wiener process by the canonical Banach-valued Wiener process. Hence we only state the lemma here and omit the proof. Although the following lemma is stated in a special form for our need in this section, it is also valid for M-type 2 Banach spaces and differential operators with Ci-smooth coefficients.

Lemma 3.7. Let E be a Ct-smooth Banach space endowed with an equivalent 2-smooth norm 1 ) . 1) . Suppose that the coeficients of H, satisfy conditions in Theorem 3.6. Then the following two Cauchy problems for parabolic equations:

and

have unique strong solutions u ( t , .) E Ci and v ( t , .) E Ci for any intial data u E C i and u E Ci . The solutions determine the positivity preserving semigroups T,.(t) and T$( t ) (t > 0) in Ci:

u ( t , .) = TP( t )u ( . ) , v ( t , .) = TT+(t)v(.)

Moreover T,(t) and T:(t) leave Ci invariant. In fact T,(t) and T;(t) can be defined by

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5


where ( l ( t ) and v:(t) are solutions of the stochastic differential equations on E

and

(Wt is the cylindrical Wiener process on H defined on some complete probability space (R, 3, P). )

Let us set

K , o = Hrlc;, Hr,p = H r l { f t - ~ ~ ; ~ 7 f c ~ ~ ) ( 1 5 P < m)

Similarly we can define the operators H$ and H A . Clearly H : , ~ C H , , and H:; C H&. Consider H:,, as an operator acting in LP to itself; denote it by H:,,, for 1 5 p < m. It is easy to see that

We shall denote the closure of by H,,o,p which is the minimal operator in LJ' associated with H,. Also Hr,,,, = H , , is called the maximal operator in Lp associated with H,. For convenience, we use T!( t ) to denote the semigroups T,(t) or T:(t). We shall use ( ( u ( ( , to denote the LP-norm of u in U ( E , p ) f o r 1 < p < + m .

Lemma 3.8. Each operator in the semigroup T!( t ) is bounded as an operator from L" n L2 -+ Lm n L2. Proof. We only prove the conclusion for the semigroup T, ( t ) . For T,i ( t ) the proof is the same. Since the semigroup T,(t) preserves the positivity of functions from Ct, we can suppose u is real. By applying Lemma 3.7, we have for ut = TT( t )u and a > 0 that

Multiplying (3.5) by e-"%, and then taking integration with respect to the

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5

1032 LONG

measure ds x dp on [0, t ] x E , we get

1 - / [e-2"tlut2 - Iu2]dp = - l t [ e - 2 a r I (-usHIus + au:)ddds . (3.6) 2 E E

It is easy to see that

H,(F'u3) = H,(F:)U; + F,?H,u! + (a*dF,?, a*du;)# = H,F: . u: + F:[~u,H,u, + (u*du,, o fdu, ) f i ] + ( a * d ~ , ? , o*du:)jj = H,F,? . u: + 2uSH,uS + F;(CT*~U, , ofduZL,)jj + (u*~F,?, o*du:)#.

From the symmetrizability of (H,, C z ) on L 2 ( E , p) , we have

0 = p , ( F : u : ) d P

= L H,F:. u: + 2 L u s H , u , d p + F:(o*du,,a*du,)fidp

+ L(o*dF,?, o*du:)#dp,

from which follows

1 L u s ~ , u , d p = - / H,F,? u:dp - - F:(o*du,, o * d ~ , ) ~ d p . (3 .7 ) 2 E 2 E 1

Combining (3 .6 ) and (3.7), we get

1 - / [e-2atut12 - Iu12]dp 2 E

1 = - l t { e z o s /&H,F: . u: + -F:(o*du,, o*du,)fi + au:]dp}ds

2 1 5 - Lt{e-'""~(- I H,') u: + au:]dp}ds.

Taking CY = llH,F:11,, we obtain

/E(e-2"tlut12 - Iul2)dp 5 0,

i.e.

IlTT(t)~ll2 I e"'llull2. (3.8)

Thus T,(t) is bounded from Cz -+ L2. Since C; is dense in L2 , T,(t) can be extended uniquely to a bounded operator from L2 -+ L2. This completes the proof.

Let us denote the semigroup acting from L2 -+ L2 by TT,2(t). Correspond- ingly for the operator H: we have a semigroup T$(t ) . We aim to show that the adjoint of T,,z(t) is T&(t) . For this, we need several lemmas.

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5

DIRlCHLET OPERATORS ON CERTAIN BANACH SPACES 1033

Lemma 3.9. T,,2(t) and T&(t) are Co-semigroups in L2. If G,,2 and G:, are the infinitesimal generators of T,,2(t) and T:2(t) respectively, then HT,0,2 G Gr,2 and H&,2 C G&. Proof. For u E Ci , we apply Ito's formula (cf.[14]) to u((;(t)) and take expectation to get

By taking the L2-norm of the both sides of (3.9) and using (3.8), we get

sup llTr,2(s)u - u/I2 5 teatllHPullz -+ 0 as t -+ 0. (3.10) O<&t

Hence T,,2(t) is strongly continuous. From (3.9), we have for u E C i

1 1 - ( T I , 2 ( t ) ~ ( x ) - u ( x ) ) - H,u(x) = - / [Ti.2(t) - IIHru(x)d~. t t o

By the Co-property of T,,z(t), it follows that

which tends to zero as t + 0. Thus the closedness of G,,2 implies H,,o,~ C G,,2. The same argument can show that H,?,,, C G,'j2.

Lemma 3.10. Let B be a Banach space, T ( t ) a Co-semigroup from B + B and G its infinitesimal generator. Let H be a closed operator in B with H 5 G. Suppose 2) is a dense subset of B so that u E 2> implies that T( t )u is in the domain of H for all t 2 0. Then H = G. Proof See Lemma 2.6 in Devinatz 1231.

Lemma 3.11. T;2(t) = TG(t) Proof. For fixed r E N, we set 4, = F,+, (m E N), then (4,) c Ci , 4m = 1 on the support of the coefficients of H, and 4, + 1 as m + co. In particular for u E Cf

and letting m -+ co, i t follows that

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5

1034 LONG

Hr,0,2(4rnut) -+ Hrut

in L2. Since 4,ut -+ ut in L2 and Hr,0,2 is closed, we have

By applying Lemma 3.9 and 3.10, we conclude that

Gr,2 = H~,o,z C Hr,2. Similarly we have H&,;, = GLz C H:2. By taking adjoints in the above relation, we get

(GL2)* = (H,t,,2)* = Hr,2. Therefore (G:2)* has a dense domain and from the theory of adjoint semigroups, it follows that Hr,2 is the infinitesimal generator of (T&(t))*. But GrI2 C Hr,2, and so the fact that they are infinitesimal generator of semigroups implies that Gr,2 = Hr,2. Hence (Trt2)* = TrS2, i.e.T,?; = T;2.

Lemma 3.12. Each operator in the semigroup {T,(t)) is bounded as an operator from Lm n LP -+ LOS n LP for 1 I p I co. Proof. T,(t) may be defined as an operator from Lm -+ Lm. In fact from the representation u(t, x ) = E[u([E(t))], it follows that

IITr(t)u(x)Ilm I eatIlullm Then combining the above estimate with (3.8) and using the Riesz-Thorin interpolation theorem (cf. Bergh and Lijfstrijm [16]) , we get

lITT(t)ullp I ea tb l (p , 2 5 P I 00. (3.12)

Let us denote by T,,p(t) the bounded mapping from LP -+ LP, 2 5 p I m . Suppose that u, v E Ci, then for 2 I p I m , T,(t)u = T,,p(t)u = Tr.p(t)u and T:(t)v = T$(t)v. From Lemma 3.11, if 2 I p < m we have

L P < T p , p ( t ) ~ l v Z L ~ = (T,,z(t)u, V ) L ~ =LP< 14, T & ( ~ ) v >LP . Thus Tj'(t)v = T;,(t)v E Lq. Since IITr,pllp = IITr';,llq, it is easy to see that T,+(t) is bounded operator from Ci -+ LQ, thus it can be uniquely extended to a bounded operator acting from LQ + LQ which coincides with T,?(t) on Lm n Lq. In the same way, we can show that T,(t) can be extended to be a bounded operator from L4 + Lq. In fact the above result is still valid for q = 1. If U E Ci , then

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5


Pointwisely we have (T , ( t )u(x) ( 4 + IT, ( t ) u ( x ) ( and ( u ( x ) ( 4 -t lu(x) 1 as q -+ 1. Since Ju(x)14 is bounded uniformly in q in a neighborhood of q = 1, we may use Fatou's Lemma and Lebesgue's dominated convergence theorem to get

JE I T . ( ~ ) U I ~ P = /E3i3 I T T ( ~ ) u I ' ~ P

5 q-+l l i m ~ l T r ( t ) u J q d r

lim em' IE lulqdp 4-1


We shall denote the extensions of { T r ( t ) ) and {T?( t ) ) acting from LP -+

by {Tr ,p ( t ) ) and {T$( t )} . From the proof of Lemma 3.12, we obviously have the following corollary.

Corollary 3.13. (Tr,p(t))* = T,lb(t), 1 < p < ca.

Lemma 3.14. {TT,p( t ) ) and {T:p(t)) are Go-semzgroups from LP -+ L P . If G , , and G:p are the correspondzng znfinzteszmal generators, then H r , ~ , p 2 G r , and H & , c_ q p . Proof. We can prove tliis lemma by using the same argument as in the proof of Lemma 3.9, the only difference being that we use the P norm instead of the L2 norm.

Lemma 3.15. Hr,o,p = GT,P = Hr,P. Proof. As in the proof of Lemma 3.11, it is easy to get

G,& = H&,, c_ H&, (1 5 9 < m). (3.14)

Taking adjoints in (3.14), we have

Thus from Corollary 3.13 and the theory of adjoint semigroups, we obtain

Hr,p = (G:¶)* = Gr,p. (3.15)

Combining (3.13) and (3.15), we complete the proof.

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5

1036 LONG

Now we turn to the proof of our main Theorem 3.6. Proof of Theorem 3.6. If u E Cb, according to the representation of ut we have

ITr,pu(x) I I Tr,p(t) IuI (x). (3.16)

Since Cb is dense in LP, (3.16) are still true for all of LP (1 I p < m ) . NOW suppose u and H,u belong to LP (1 < p < m ) in the sense of ((2:)'. Then there exists g E LP such that Vv E C:

i.e, u E V ( H ,,,,,) = V(H;) and H, ,,,, u = g. Let us show that u E V(Hr,o,p) for any T E N. According to the definition of the operator H, acting on generalized functions, we have

i.e.Hru = F:Hpu E LP. By Lemma 3.15 and the definition of Hr,p, we get

Hence

By Corollary 3.13, we know that T;,(t) = TptP(t). Therefore if v E C: we have

where HTIuI is in the sense of generalized functions in (C;)*. On the other hand,u E Z ~ ( H , , ~ ) . Hence

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5


If v 2 0, then combining (3.18), (3.19) with (3.17), we conclude that

Finally we must pass to the limit in (3.20) when r -+ cm. Xote that H,luJ -+ H,juj as r + cm in the sense of generalized functions in (Ci)*,i.e. Vv E C i , we have

LP < H T ( u ( , v > . ~ ~ = ~ ~ < ( u ( , H T ~ v > L B + L~ < (U( ,H,V>LP

as T + cm, which follows from the fact that H,+v + H,v in Lq. Moreover,

H;,u = F,?H,U -+ H,u in Lp.

Hence by letting r -+ m, (3.20) becomes the inequality

If we use the test function space C i , then we need more smoothness on the Banach space E. In this case we can also obtain the corresponding Kato's inequality in the L2-setting and in the sense of generalized function in (Ci)* as we now describe. We assume that E satisfies the following hypothesis (HI) there exists a sequence of Ct-bump functions { F r ( . ) ) r c ~ such that (1) 0 5 FT < 1 and

1 for z E {x E E; ljxlj 5 r ) F.(x) = { 0 for x E {x E E ; llxll 2 T + 1)

(2) there exists a constant K > 0 such that Vx E E

Theorem 3.16. Let E be a Banach space satisfying ( H I ) endowed with an equivalent 2-smooth norm ( 1 . / I . The coeficients of H, satisfy the conditions in Th,eorem 3.6. If u and H,u belong to L2, then Kato's inequality holds in the seme of generalized functions in (C;)"

Re[(sgnii)H,u] 5 H,JuJ.

Proof. We can follow the same procedure as in the proof of Theorem 3.6 to prove our Theorem 3.16. In fact the proof is somewhat simple. The reason we need E to satisfy ( H I ) is that if we want, for u E C;,

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5

1038 LONG

HTu + HPu and H:u -+ H,u in L2,

we require that

Remark 3.17. We can provide some Banach spaces which satisfy (Hl). Let Cp(R, u) be a Banach space for a finite positive measure space ( 0 , u) with the standard norm llxllp = (Jn I X ( W ) I P V ( ~ W ) ) ; . If p = 2 or p 2 4, then CP(0, u) is a Ct-smooth Banach space satisfying (HI). In fact, from the basic cal- culation in Bonic and Frampton 1151, we know that $(x) = 11x11, E C 4 for x E E\{O) and there exists a constant Kp such that

We define a sequence of Ci-functions 8, : R+ + [O, 11, T E N by

BT(s)= 1 ,VsE [O,r] and OT(s)=O,VsE [ ~ + l , + c o ) ; B , + ~ ( s + l ) =B,(s).

Now if we set FT(.) = BT(1l . Itp) for each r E N, then obviously {F,(.)) is a sequence of Ci-bump functions on CP(0, u) satisfying (HI ) . Indeed, if we set = 8 then Vx E CP(0, u)

4 ESSENTIAL SELF-ADJOINTNESS OF DIRICHLET OPERATORS

In this section we consider the essential self-adjointness of Dirichlet operators on certain smooth Banach spaces. First we apply Kato's inequality to prove the essential self-adjointness of H, with non-constant diffusion part defined as in the previous section. For this we need the following well-known criterion for essential selfadjointness of differential operators (cf. Theorem 1 of Section 11.3 in Hellwig 1281).

Lemma 4.1. Let X be a separable Hilbert space and L a dzfferentzal operator with dense domain V ( L ) c X. Assume that

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5


(1) (L, V(L)) i s symmetric on X , (2) LV(L) i s a dense subset of X , and (3 ) for all u E V(L), there exists a positive constant cr such that llLulIx 2

Q I I u I I x . T h e n (L, V(L)) i s essentially self-adjoint.

Theorem 4.2. Let the assumptions of Theorem 3.6 be fulfilled. If there exists a constant C > 0 such that llH,F:112 5 C for all r E N , then the Dirichlet operator H, zs essentially self-adjoint with essential domazn Ci. Proof. It is sufficient to prove the essential self-adjointness of H, - 1. Now we check conditions (I), (2) and (3) of Lemma 4.1. Of these, (1) is obvi- ous. For (3), by using the symmetrizability of (H,, Ci) on L2(E, p ) and the Cauchy-Schwartz inequality, we have for any u E Ci

11~112 s ll(H, - l h l l z . For (2), We only need to show that (H, - 1)u = 0 implies that u = 0 (in the sense of generalized functions in (Ci)* ). From (H, - 1)u = 0, it follows that H,u = u E L2. By applying Kato's inequality, we obtain

H,IuI 1 Re[(sgnfi)H,u] = Re[(sgn~)u] = lul. (4.1)

Hence

(1 - H,)luI I 0. (4.2)

From Lemma 3.5 we know that C i is dense in L2, then for I u I E L2 and any given E > 0 there exists v E C i such that

E ll(lul - v)ll2 < z.

On the other hand, there exists TO E N such that suppv C { x E E; llxll 5 TO),

which implies that if T 1 TO

Therefore when T 2 1-0 we have

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5

LONG

which implies that

( 1 ~ 1 , H ~ F : ) -+ 0 as r + m.

Now it is easy to see that

as r + cc .From this fact and (4.2), it follows that

l l 4 l 5 01

which implies that u = 0. This completes the proof.

Similarly by applying Theorem 3.16, we have the following result.

Theorem 4.3. Under the conditions of Theorem 3.16, the operator H,, is essentially self-adjoint with essential domain Cz.

In the rest of this section we shall provide an approximation criterion for essential self-adjointness of Dirichlet operators H,, with constant diffusion part on general class of Banach spaces, i.e. M-type 2 Banach spaces. For this we need to use the classical notion of semi inner-product on Banach spaces.

Definition 4.4. A semi inner-product on complex or real vector spaces E is a mapping [ , I : E x E -+ C(or R) such that (2) [x + Y, 4 = 1x1 4 + [Y, 21 (ii) [ X X , ~ ] =X[x,y],X E C (or R) (iii) [x, X I > 0 for x # 0 6.1 I I x , y1I2 i 15, xI[Y, Y I for X, Y E E. Such a vector space E is called a semi inner-product space (s.2.p.s).

Remark 4.5. (1) s.i.p.s E is a normed space. In fact we can define the norm

I I . I I on E by IIxII = [x,xI+. (2) Any Banach space E can be made into a s.i.p.s. According to the Hahn- Banach Theorem for each x E E there exists at least one x* E E* such that < x,x* >= Ijxlj2. Then clearly [x, y] =< x, y* > for each x, y E E defines a semi inner-product on E. In this case we call [ , I a compatible semi inner- product with respect to norm I / . 1 1 on E.

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5


(3) The concept s i p s is introduced by Lumer [38]. Then Lumer, Phillips and Kato etc dealt with the evolution equations on Banach space E with a compatible semi inner-product via dissipative operators on E.

Proposition 4.6. Every M-type 2 Banach space E admits an equivalent 2-smooth norm and an unique compatible semi inner-product [,] on E x E with respect to this equivalent norm. Proof. From Remark 2.10, we know that E is 2-smooth. We may assume the equivalent norm is 1 1 . 1 1 . Since 1 1 . 1 1 is Frkchet differentiable, from Remark 2.13 we know that J(x) is single-valued and J(x) = (JJxJJ) ' . Now we define the semi inner product [x, y] on E x E by

[x, y] =< x , y* > for any x, y E E,

where y* = (ly(lJ(y) E E*. Obviously [x , y] is the unique compatible semi inner-product on E x E with respect to 11 . 1 1 . This completes the proof.

From the Remark 3.3, if (H, 2) = (H, E ) and a = i , we can define the symmetric differentia1 operator H, by

1 1 H,u = -AHU + - E < P(x), fl(x) > E * ,

2 2 where AH is Gross's Laplacian and ,B : E -, E is the vector logarithmic derivative of measure p satisfying (pl) . We state the approximation criterion for essential self-adjointness of H, on L2(E, p) as follows Theorem 4.7. Let E be a M-type 2 Banach space endowed with a 2-smooth norm 1) . 1 ) and a compatible semi inner-product [ , ] on E x E. Suppose that there exists a sequence of mappings {bn)nEN from E into E satisfying (i) V n E N , b, E Ct(E , E); (ii) - b.11 -+ 0 as n -+ cc in L2(E,p) ; (iii) there exists an absolute constant C E R such that

[&(x)h, h] 5 CI)hJJ2,Vh E E and n E N,

where bk(x) E C(E), x E E denotes the Fre'chet derivative of the mapping b,. Then H, is essentially self-adjoint in L2(E, p) with essential domain C;.

Before proceeding the proof of the above theorem, we give several lemmas which are crucial. Now we define a sequence of operators {Hn)nEN as follows

Lemma 4.8. Suppose E satisfies the condition of Theorem 4.7. Also we assume that A(s) is in C(E) for every s E R+ and there exzsts a constant

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5

1042 LONG

C E R such that

[A(s )h , h] 5 Cllh112 for all s E R+.

Let ~ ( t ) be a strong solution of the following equation

u ( t ) = h + it A ( s ) u ( s ) ~ s , t 2 0 , h E E. (4.3)

Then

lI4t)ll 5 IlhlleCt. (4.4)

Proof. Since A(s )u(s ) is Bochner-integrable, according to the Bochner The- orem u ( s ) is strong differentiable a.e.s 2 0. Set l ( x ) = l/xl/. Since 1 1 . 1 1 is uniformly smooth, thus J ( x ) is single-valued and J ( x ) = l i ( x ) . Therefore 11~(s)1/~ is differentiable in s a.e. s 2 0 and

which is equivalent to

I I U ( ~ ) I I ~ I I ~ I I ~ + 2 c i t I I U ( S ) I I ~ ~ S . (4.5)

By using Gronwall's inequality, we get

llu(t)ll 5 IlhlleCt Thus we complete the proof.

Now let us consider the following Cauchy problem

where u E C z ( E ) . From Lemma 3.7, we know that the above Cauchy problem has a unique strong solution in the class of Cz. Moreover u,(t) can be represented by using the solution of the following stochastic differential equations

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5

DIRlCHLET OPERATORS ON CERTAIN BANACH SPACES

un(t,x) = E[u(tn,x(t))], x E E , t E R+. We give the following estimate for un ( t , x). Lemma 4.9. For u E C:(E), un(t, .) E C;(E) and we have

Proof. For any n E N , the process J,,,(t) is differentiable with respect to x E E a.s. (cf.[lO]) and ~ : , ~ ( t ) satisfies the equation

1 t <.,,(t)h = h + 1 &(4,x(s ) ) i ,x (s )hds . (4.9)

Set V&(t) = <:,,(t)h. By using assumption (iii) and Lemma 4.8, we have

Ilv:x(t)ll 5 11hlle4~' Now it is easy to calculate that


Now we can turn to the proof of Theorem 4.7. Proof of Theorem 4.7. We apply the following general parabolic criterion of essential self-adjointness (cf. Theorem 6.13 of Chapter 2 in Berezanskii [IS]). If the Cauchy problems (4.6) have unique strong solutions u, ( t , x ) and u,(t, .) belong to V(H,) for n E N , and

1' l l(Hp - Hn)un(t)l/2dt + 0 as n -+ m,

then the operator (H,, V(H,)) is essentially self-adjoint. Since D(H,) = Cf and from Lemma 3.6 we know that the Cauchy problems (4.6) have unique strong solutions un ( t , x ) with un ( t , .) in C:, if we can prove the convergence

then the operator H, is essentially self-adjoint with essential domain C:. In fact, by using the estimate (4.8) and assumption (ii), we have

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5

as n + m. This completes the proof.

If we further suppose that there exists an increasing sequence of finite dimensional projections {P,) from E into E" such that Pn converges strongly to the identity operator both in E and H. Then by using the finite dimensional approximation argument as in Albeverio, Kondratiev and Rockner [6] we can prove the following theorem:

Theorem 4.10. Under the above assumption on E and the conditions of Theorem 4.7, H, is essentially self-adjoint with essential domain FC," on L 2 ( E , P ) .

Remark 4.11. Under similar abstract setting of [6] by using the triple E* C H C E instead of the rigged Hilbert spaces H+ c H c H-, it is possible to prove the same results as in [6] via the classical notion of semi inner-product.

ACKNOWLEDGMENTS

This paper was completed under the delicate supervision of Prof. K. D. Elworthy. The author would like to express his sincere gratitude to him. The financial support of Overseas Research Scholarship and Ella Lam Fellowship is gratefully acknowledged.

References

[1] E. Acosta, On the essential self-adjointness of Dirichlet operators on group-valued path spaces, Proc. Amer. Math. Soc. 122 (1994), 581-590.

[2] S. Aida, Sobolev spaces over loop groups, J. Funct. Anal. 127 (1995), 155-172.

[3] S. Albeverio and R. HBegh-Krohn, Hunt processes and analytic potential theory on rigged Hilbert spaces, Ann. Inst. Henri Poincari, Sect. B. 13(1977), 269-291.

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5


S. Albeverio, G. Yu. Kondratiev and M. Rockner, An approximate criterium of essential self-adjointness of Dirichlet operators, Potential Anal. 1(1992), 307-317.

S. Albeverio, G.Yu. Kondratiev and M. Rockner, Addendum to the paper "An approximate criterium of essential self-adjointness of Dirichlet operators ", Potential Anal. 2 (1993), 195-198.

S. Albeverio, G.Yu. Kondratiev and M. Rockner, Dirichlet operators via stochastic analysis, J. Funct. Anal. 128 (1995), 102-138.

S. Albeverio, Z.M. Ma and M. Rockner, Partition of unity in Sobolev spaces over infinite dimensional state spaces, Bielefeld Preprint 95-009, 1995.

V.I. Averbuh, O.G. Smolyanov and S.V. Fomin, Generalized functions and differential equations in linear spaces, Trans. Moscow Math. Soc. 24 (1971), 140-184.

V. Barbu, Nonlinear Semigroups and Dzfferential Equations in Banach Space, Noordhoff International Publishing, Leydon the Netherlands, 1976.

Ja.1. Belopolskaya and Yu.L. Daletskii, Diffusion processes in smooth Banach spaces and manifolds, Trans. Moscow. Math. Soc. 37 (1978), 113-150.

Ja.1. Belopolskaya and Yu.L. Daletskii, Stochastic Equatlon and Differ- ential Geometry, Kluwer, Academic Publisher, 1989.

D.R. Bell, The Malliauin Calculus, Pitman Monographs and Surveys in Pure and Applied Mathematics 34, Longman Scientific and Technical, 1987.

[13] Yu.M. Berezanskii, Self-adjoint Operators in Spaces of Functions of In- finitely Many Variables, Transl. Math. Monography, vol63, Amer. Math. Soc., Providence RI, 1986.

[14] Z. Brzezniak and K.D. Elworthy, Stochastic differential equation on Ba- nach manifolds, Warwick preprint, 1996.

[15] R. Bonic and J. Frampton, Smooth functions on Banach manifolds, J. Math. Mech. 15 (1966), 877-898.

[16] J. Bergh and J. Lofstrom, Interpolation Spaces, Grundlehren der math- ematicschen Wissenschaften 223, Springer-Verlag, 1976.

1171 V.I. Bogachev and M. Rockner, Regularity of invariant measures on

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5

1046 LONG

finite and infinite dimensional spaces and applications, J. Funct. Anal. 133 (1995), 168-223.

[18] V.I. Bogachev and O.G. Smolyanov, Analytic properties of infinite dimensional distributions, Russian Math. Surveys,45 (1990), 1-104.

[19] R. Carmona, Tensor product of Gaussian measure, in " Vector Space Measures and Application I ",pp96-124, Lecture Notes in Math. 644, Springer-Verlag, 1978.

[20] H.L. Cycon, R.G. Froese, W. Kirsh and B. Simon, Schrodinger Operators with Application to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics, Springer-Verlag, Berlin, Heidelberg, 1987.

[21] Yu.L. Daletskii, Infinite dimensional elliptic operators and parabolic equation connected with them, Russian Math. Surveys 22 (1967), 1-53.

[22] Yu.L. Daletskii, On the self-adjointness and maximal dissipativity of differential operators for functions of an infinite dimensional argument, Sovaet Math. Dokl. 1 7 (1976), 498-502.

[23] A. Devinatz, On an inequality of Tosio Kato for degenerate-elliptic opeartors, J. Funct. Anal. 32 (1979), 312-335.

[24] Yu.L. Daletskii and S.V. Fomin, Measures and Differential Equations i n Infinite Dimensional Spaces, Kluwer, Dordrecht, 1991.

1251 R. Deville, G. Godefroy and V. Zizler, Smoothness and Renorming i n Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64, Longman Scientific and Technical, 1993.

[26] K.D. Elworthy and Z.M. Ma, Vector fields on mapping spaces and related Dirichlet forms and diffusions, Warwick Preprints: 24/1996.

[27] V. Goodman, A divergence theorem for Hilbert space, Trans. Amer. Math. Soc. 164 (1972), 411-426.

[28] G. Hellwig, Differential Operators of Mathematzcal Physics, Addison- Wesley Publishing Company, 1967.

[29] T . Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, 19(1967), 508-520.

[30] T. Kato, Schrodinger operators with singular potentials, Israel J. Math. 13 (1973), 136-148.

[31] Yu.G. Kondratiev, The Kato's inequality for second quantization operators, Ukrazn. Math. J. 35 (1983), 753-758.

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5


[32] Yu.G. Kondratiev, Dirichlet operators and smoothness of solutions of elliptic equations, Soviet Math. Dokl. 31 (1985), 461-464.

[33] Yu.G. Kondratiev and T.V. Tsycalenko, Infinite dimensional Dirichlet operators (I): Essential self-adjointness and associated elliptic equations, Potential Anal. 2 (1993), 1-22.

1341 H.H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Math., Vol 463, Springer-Verlag, Berlin, New York, 1975.

[35] S. Lang, Introduction to Dzfferential Manifolds, New York, Interscience Publisher 1962.

[36] V.A. Liskevich and Yu.A. Semennov, Dirichlet operators, a priori esti- mates, uniqueness problem, J. Funct. Anal.109 (1992), 199-213.

[37] H.W. Long, Necessary and sufficient conditions for the symmetrizability of differential operators with non-constant diffusion part on infinite dimensional state spaces, In preparation.

(381 G. Lumer, Semi inner-product spaces, Trans. Amer. Math. Soc. 100 (1961), 29-43.

[39] N.V. Norin, Stochastic integrals and differentiable measures, Theory Probab. Appl. 32 (1987), 107-116.

[40] G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), 326-350.

[41] I. Shigekawa, An example of regular (r,p)-capacity and essential self- adjointness of a diffusion operator in infinite dimension, J. Math. Kyoto. Univ. 35 (1995) ,639-651.

[42] A.V. Skorohod, Integration an Hilbert Space, Springer- Verlag, New York, Heidelberg, 1974.

[43] S.Q. Song, Markov uniqueness and essential self-adjointness of perturbed Ornstein-Uhlenbeck operators, Osaka J. Math. 32 (1995),823-832.

[44] O.G. Smolyanov and H.V. Weizsacker, Differentiable families of measures, J. Funct. Anal. 118(1993), 454-476.

[45] J.C. Wells, Differential functions on co, Bull. Amer. Math. Soc. 7 5 (1969), 117-118.

[46] J. C. Wells, Differentiable functions on Banach spaces with Lipschitz derivatives, J. Di8. Geom. 8 (1973), 135-152.

(471 N. Wielens, On essential self-adjointness of generalized Schrodinger operators, J. Funct. Anal. 6 1 (1985), 98-115.

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

2:17

18

May

201

5

dirichlet operators on certain banach spaces kato's …math.fau.edu/long/long1998saa.pdfcal...

Documents