OPERATOR-VALUED FUNCTION SPACE INTEGRALS VIA CONDITIONAL INTEGRALS ON AN ANALOGUE WIENER SPACE II

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1 Bull. Korean Math. Soc. 53 (16), No. 3, pp pissn: / eissn: OPERATOR-VAUED FUNCTION SPACE INTEGRAS VIA CONDITIONA INTEGRAS ON AN ANAOGUE WIENER SPACE II Dong Hyun Cho Abstract. In the present paper, using a simple formula for the conditional expectations given a generalized conditioning function over an analogue of vector-valued Wiener space, we prove that the analytic operatorvalued Feynman integrals of certain classes of functions over the space can be expressed by the conditional analytic Feynman integrals of the functions. We then provide the conditional analytic Feynman integrals of several functions which are the kernels of the analytic operator-valued Feynman integrals. 1. Introduction et r be a positive integer and let C[,t] r [7] denote the r-dimensional Wiener space. On the space C r [,t] Cameron and Storvick [1] introduced a very general analytic operator-valuedfunction space Feynman integral Jq an(f), which mapped an (R r )-function ψ into an (R r )-function Jq an (F)ψ. In [, 1] the existence of the analytic operator-valued Feynman integral Jq an (F) as an operator from 1 (R) to (R) was studied, and Chung, Park and Skoug [7] showed that it can be expressed by the conditional analytic Feynman integral of F. Further work extending the above ( 1, )-theory with the conditional analytic Feynman integrals was studied by the author [5] over the space (C r [,t],wϕ) r [9, 11] of the continuous R r -valued paths on [,t] which generalizes the space C r [,t]. In fact the author [4] introduced the conditional Wiener integral over C r [,t] and derived a simple formula for the conditional Wiener integral with the conditioning function X n : C r [,t] R (n+1)r given by (1) X n (x) = (x(t ),x(t 1 ),...,x(t n )), Received June 1, 15; Revised August 19, Mathematics Subject Classification. Primary 6H5; Secondary 8C, 46T1. Key words and phrases. analogue of Wiener measure, conditional analytic Feynman integral, conditional analytic Wiener integral, operator-valued Feynman integral, simple formula for conditional expectation, Wiener space. This work was supported by Kyonggi University Research Grant c 16 Korean Mathematical Society

2 94 DONG HYUN CHO where = t < t 1 < < t n = t, which calculates directly the conditional Wiener integral in terms of the ordinary non-conditional Wiener integral. Applying this simple formula to a certain function F defined on C r [,t] with the conditioning function X 1 : C r [,t] R r defined by X 1 (x) = (x(),x(t)), he [5] could express the analytic operator-valued Feynman integral Jq an(f) : 1 (R r ) (R r ) in terms of the conditional analytic Feynman integral E anfq [F X 1 ] of F given X 1. In the present paper we further develop the concepts in [5] with more generalized conditioning function X n (n 1) given by (1). For the conditioning function X n we proceed to express the analytic ( 1, )-operator valued Feynman integrals in terms of the conditional analytic Feynman wϕ-integrals. r In fact we establish that for certain functions F on C r [,t] and for a nonzero real q, the analytic operator-valued Feynman integral Jq an (F) exists as an element of ( 1 (R r ), (R r )), the space of the bounded linear operators from 1 (R r ) to (R r ), and it is given by the formula (J an q (F)ψ)(ξ) = ( iq) r E anfq [F X n ](ξ,ξ 1,...,ξ n )ψ(ξ n )Ψ( iq,ξ R (n+1)r ξ)w r ( iq, τ n,(ξ,ξ 1,...,ξ n ))dm r (ξ )dm r (ξ 1 ) dm r (ξ n ) for ψ 1 (R r ) and m r -a.e. ξ Rr, where m r is the ebesgue measure over R r, W r is given by W r ( iq, τ n,(ξ,ξ 1,...,ξ n )) [ n ]r { q iq = exp πi(t j t j 1 ) n ξ j ξ j 1 } R r t j t j 1 and Ψ is the analytic extension of the probability density of ϕ r. Thus J an q (F) can be interpreted as an integral operator with the kernel ( iq) r E anf q [F X n ](ξ,ξ 1,...,ξ n )Ψ( iq,ξ ξ)w r ( iq, τ n,(ξ,ξ 1,...,ξ n )). We then provide the conditional analytic Feynman w r ϕ-integral for the cylinder functions which are important in quantum mechanics and Feynman integration theoriesthemselves. Wenotethatifϕ r = δ,thediracmeasureconcentratedat R r, then C r [,t] is identified with the r-dimensional Wiener space C r [,t] so that our worksin this paper generalize those of [7] when n = 1. Furthermore if n = 1, then most results of this paper can be reduced to those in [5], that is, the works in this paper also extend the results in the same reference.. An analogue of the r-dimensional Wiener space Throughout this paper let C, C + and C + denote the sets of the complex numbers, the complex numbers with positive real parts and the nonzero complex numbers with nonnegative real parts, respectively. Furthermore let m

3 OPERATOR-VAUED FUNCTION SPACE INTEGRAS 95 denote the ebesgue measure on the Borel class B(R) of R. The dot product on the r-dimensional Euclidean space R r is denoted by, R r. For a positive real t let C = C[,t] be the space of all real-valued continuous functions on the closed interval [, t] with the supremum norm. et (C[,t],B(C[,t]),w ϕ ) denote the analogue of Wiener space associated with the probability measure ϕ [9, 11], where ϕ is a probability measure on B(R). et C r = C r [,t] be the product space of C[,t] with the product measure wϕ r. Since C[,t] is a separable Banach space, B(C r [,t]) = r B(C[,t]). This probability measure space (C r [,t],b(c r [,t]),wϕ r ) is called an analogue of the r-dimensional Wiener space. For v in [,t] and x in C[,t] let (v,x) denote the Paley-Wiener-Zygmundintegral of v according to x [9] and let, denote the inner product over [,t]. emma.1 ([9, emma.1]). If f : R n+1 C is a Borel measurable function, then = where () C R f(x(t ),x(t 1 ),...,x(t n ))dw ϕ (x) R n f(u,u 1,...,u n )W 1 (1, t n,(u,u 1,...,u n ))dm n (u 1,...,u n )dϕ(u ), = W r (λ, t n,(u,u 1,...,u n )) [ n ]r { λ exp λ π(t j t j 1 ) n u j u j 1 } R r t j t j 1 for r N, λ C +, t n = (t,t 1,...,t n ) with = t < t 1 < < t n t, (u,u 1,...,u n ) R (n+1)r, and = means that if either side exists, then both sides exist and they are equal. Now we introduce a useful lemma which plays a key role in the proof of Theorem 3.. The proof of it is similar to the proof of emma 3.4 in [5]. emma.. For t n = (t,t 1,...,t n ) with = t < t 1 < < t n t, λ > and ξ R r, let X λ,ξ n : C r [,t] R (n+1)r be the function given by X λ,ξ n (x) = (λ 1 x(t )+ξ,λ 1 x(t1 )+ξ,...,λ 1 x(tn )+ξ). Furthermore let P X λ,ξ be the probability distribution of X λ,ξ n n on the Borel class B(R (n+1)r ) of R (n+1)r and suppose that ϕ r is absolutely continuous with respect to the ebesgue measure m r. Then P X λ,ξ n dp X λ,ξ n dm (n+1)r m (n+1)r and (ξ,ξ 1,...,ξ n ) = λ r Wr (λ, t n,(ξ,ξ 1,...,ξ n )) dϕr dm r (λ 1 (ξ ξ)) for m (n+1)r -a.e. (ξ,ξ 1,...,ξ n ) R (n+1)r, where W r is given by ().

4 96 DONG HYUN CHO 3. A simple formula for conditional wϕ r -integrals and the operator-valued function space integrals In this section we introduce a simple formula for an analogue of the conditional Wiener integralsoverc r [,t] with a vector-valuedconditioning function. et F : C r [,t] C be integrable and let X be a random vector on C r [,t] assuming that the value space of X is a normed space with the Borel σ-algebra. Then we have the conditional expectation E[F X] of F given X from a well known probability theory. Furthermore there exists a P X -integrable complexvalued function ψ on the value space of X such that E[F X](x) = (ψ X)(x) for wϕ r-a.e. x Cr [,t], where P X is the probability distribution of X. The function ψ is called the conditional wϕ r -integral of F given X and it is also denoted by E[F X]. Throughout this paper, let τ n = (t,t 1,...,t n ) be given with = t < t 1 < < t n = t. For any x in C r [,t] define the polygonal function [x] by n ( tj s [x](s) = χ (tj 1,t j](s) x(t j 1 )+ s t ) j 1 (3) x(t j ) t j t j 1 t j t j 1 +χ {t}(s)x(t ) fors [,t], whereχ (tj 1,t j] andχ {t} denotetheindicatorfunctions. Similarly, for ξ n = (ξ,ξ 1,..., ξ n ) R (n+1)r, define the polygonal function [ ξ n ] by (3), where x(t j ) is replaced by ξ j for j =,1,...,n. In the following theorem we introduce a simple formula for the conditional w r ϕ-integrals on C r [,t] [4]. Theorem 3.1. et F : C r [,t] C be integrable and X n : C r [,t] R (n+1)r be given by (4) Then for P Xn -a.e. ξ n R (n+1)r, X n (x) = (x(t ),x(t 1 ),...,x(t n )). E[F X n ]( ξ n ) = E[F(x [x]+[ ξ n ])], where P Xn is the probability distribution of X n on (R (n+1)r,b(r (n+1)r )) and the expectation is taken over the variable x. (5) and (6) et F : C r [,t] C be a function. For notational convenience let X λ,ξ n (x) = X n (λ 1 x+ξ) F λ,ξ (x) = F(λ 1 x+ξ) for λ > and for ξ R r. Suppose that F λ,ξ is integrable over C r [,t]. Then, by Theorem 3.1, (7) E[F λ,ξ X λ,ξ n ]( ξ n ) = E[F(λ 1 (x [x])+[ ξn ])]

5 OPERATOR-VAUED FUNCTION SPACE INTEGRAS 97 for P X λ,ξ -a.e. ξn = (ξ n,ξ 1,...,ξ n ) R (n+1)r, where P X λ,ξ n distribution of Xn λ,ξ on (R (n+1)r,b(r (n+1)r )). et (8) (K λ (F))( ξ n ) = E[F(λ 1 (x [x])+[ ξn ])]. is the probability If (K λ (F))( ξ n ) has the analytic extension J λ (F)( ξ n ) on C + as a function of λ, then it is called the conditional analytic Wiener w r ϕ -integral of F given X n with parameter λ and denoted by [F X n ]( ξ n ) = J λ(f)( ξ n ) for ξ n R (n+1)r. Moreover, if for a nonzero real q, [F X n ]( ξ n ) has the limit as λ approaches to iq through C +, then it is called the conditional analytic Feynman w r ϕ -integral of F given X n with parameter q and denoted by E anfq [F X n ]( ξ n ) = lim λ iq Eanw λ [F X n ]( ξ n ). et F : C r [,t] C be a measurable function. For any λ >, ψ in 1 (R r ) and ξ in R r, let ψ λ,ξ t (x) = ψ(λ 1 x(t)+ξ) and (I λ (F)ψ)(ξ) = F λ,ξ (x)ψ λ,ξ t (x)dwϕ r (x), C r where F λ,ξ (x) is given by (6). If I λ (F)ψ is in (R r ) as a function of ξ and if the correspondence ψ I λ (F)ψ gives an element of ( 1 (R r ), (R r )), we say that the operator-valued function space integral I λ (F) exists. Next suppose that there exists an -valued function which is weakly analytic in C + and agrees with I λ (F) on (, ). Then this -valued function is denoted by Iλ an(f) and is called the analytic operator-valued Wiener wr ϕ-integral of F associated with parameter λ. Finally, for a nonzero real q, suppose that there exists an operator Jq an(f) in such that for every ψ in 1(R r ), Iλ an(f)ψ convergesweaklytojq an (F)ψ asλapproachesto iq throughc +. ThenJq an (F) iscalledtheanalyticoperator-valuedfeynmanwϕ r -integraloff withparameter q. We can take ψ to be Borel measurable [8], and the weak limit and the weak analyticity are based on the weak topology on (R r ) induced by its pre-dual 1 (R r ) [, 1]. Theorem 3.. et the assumptions and notations be as given in emma. and X n be given by (4). For F : C r [,t] C suppose that [F X n ]( ξ n ) exists for λ C + and m (n+1)r -a.e. ξn R (n+1)r, and that for each bounded subset Ω of C +, there exists M Ω such that (9) [F X n ]( ξ n ) M Ω for all λ Ω and m (n+1)r -a.e. ξ n R (n+1)r. Furthermore suppose that there exists a function Ψ on C + R r satisfying the following conditions: (i) for each λ >, Ψ(λ,η) = dϕr dm r (λ 1 η) for m r -a.e. η Rr, (ii) for m r -a.e. η Rr, Ψ(λ,η) is analytic on C + as a function of λ, and

6 98 DONG HYUN CHO (iii) for each bounded subset Ω of C +, Ψ(λ,η) is bounded for all λ Ω and for m r -a.e. η Rr. Then for λ C +, the analytic operator-valued Wiener wϕ r -integral Ian λ (F) exists as an element of and is given by (1) (I an λ (F)ψ)(ξ) = λ r [F X n ](ξ,ξ 1,...,ξ n )ψ(ξ n )Ψ(λ,ξ ξ) R (n+1)r W r (λ, τ n,(ξ,ξ 1,...,ξ n ))dm r (ξ )dm r (ξ 1) dm r (ξ n) for ψ 1 (R r ) and m r -a.e. ξ Rr, where W r is given by (). In addition, suppose that n = 1 and for a nonzero real q, E anfq [F X 1 ](ξ,ξ 1 ) exists for m r-a.e. (ξ,ξ 1 ) R r. Moreover suppose that Ψ can be extended to (C + { iq}) R r with the following two additional conditions: (ii) for m r -a.e. η Rr, Ψ(λ,η) is continuous at λ = iq as a function of λ, and (iii) there exists a (Borel measurable) function Φ q 1 (R r ) satisfying Ψ(λ,η) Φ q (η) for all λ Ω ǫ and m r -a.e. η R r, where Ω ǫ = {λ C + : λ+iq < ǫ} for some ǫ >. Then the analytic operator-valued Feynman wϕ r -integral Jan q (F) exists as an element of and it is given by (11) (Jq an (F)ψ)(ξ) = ( iq) r E anfq [F X 1 ](ξ,ξ 1 )ψ(ξ 1 ) R r R r Ψ( iq,ξ ξ)w r ( iq, τ 1,(ξ,ξ 1 ))dm r (ξ )dm r (ξ 1 ). Proof. For λ >, ψ 1 (R r ) and ξ R r (I λ (F)ψ)(ξ) = E[F λ,ξ ψ λ,ξ t Xn λ,ξ ](Xn λ,ξ (x))dwϕ(x) r C r = E[F λ,ξ ψ λ,ξ t Xn λ,ξ ]( ξ n )dp ( X λ,ξ ξ R (n+1)r n n ), where Xn λ,ξ is given by (5) and P X λ,ξ is the probability distribution of X λ,ξ n n on the Borel class of R (n+1)r. For ξ n = (ξ,ξ 1,...,ξ n ) R (n+1)r E[F λ,ξ ψ λ,ξ t X λ,ξ n ]( ξ n ) = E[F(λ 1 (x [x])+[ ξn ])ψ(λ 1 (x [x])(t)+[ ξn ](t))] = (K λ (F))( ξ n )ψ(ξ n ) by (7), where K λ (F) is given by (8). By emma. (I λ (F)ψ)(ξ) = λ r (K λ (F))(ξ,ξ 1,...,ξ n )ψ(ξ n )W r (λ, τ n,(ξ,ξ 1, R (n+1)r...,ξ n )) dϕr dm r (λ 1 (ξ ξ))dm r (ξ )dm r (ξ 1) dm r (ξ n).

7 OPERATOR-VAUED FUNCTION SPACE INTEGRAS 99 Now suppose that Ψ satisfies (i), (ii) and (iii). By (i) (I λ (F)ψ)(ξ) = λ r (K λ (F))(ξ,ξ 1,...,ξ n )ψ(ξ n )Ψ(λ,ξ ξ) R (n+1)r W r (λ, τ n,(ξ,ξ 1,...,ξ n ))dm r (ξ )dm r (ξ 1 ) dm r (ξ n ). et Ω λ be a bounded subset of C + containing λ. Then for m r -a.e. ξ Rr (I λ (F)ψ)(ξ) λ r MΩλ Ψ Ωλ, ψ 1(R r ), where Ψ Ωλ, denotes the essential supremum of Ψ on Ω λ R r, so that I λ (F)ψ (R r ) and I λ (F) ( 1 (R r ), (R r )). For ψ 1 (R r ) let (Q λ (F)ψ)(ξ) be the right hand side of (1) for (λ,ξ) C + R r and let Ω be any bounded subset of C +. By the same method ( (1) (Q λ (F)ψ)(ξ) λ r λ MΩ Ψ Ω, ψ 1(R r ) Reλ for all λ Ω and m r -a.e. ξ Rr so that Q λ (F)ψ (R r ) and Q λ (F) ( 1 (R r ), (R r )) for λ C +. Using the same method as used in the proof of Theorem 3.5 in [5] we can prove that Iλ an (F) exists and Ian λ (F) = Q λ(f) for λ C +. The remainder part of the theorem follows from Theorem 3.5 in [5]. 1 etting Ψ(λ,η) = ( πα ) r exp{ λ α η R r} for α >, λ C + and η R r, we have the following corollary from Theorem 3.. Corollary 3.3. et X n be given by (4). Moreover let ϕ r be normally distributed with the mean vector R r and the nontrivial variance-covariance matrix α I r, where α > and I r is the r-dimensional identity matrix. For F : C r [,t] C suppose that [F X n ]( ξ n ) satisfies (9) in Theorem 3.. Then for λ C +, the analytic operator-valued Wiener wϕ-integral r Iλ an (F) exists as an element of and is given by (13) = (I an )nr λ (F)ψ)(ξ) ( )r λ πα [F X n ](ξ,ξ 1,...,ξ n )ψ(ξ n )W r (λ, τ n, R(n+1)r { (ξ,ξ 1,...,ξ n ))exp λ ξ ξ R r α } dm r (ξ )dm r (ξ 1 ) dm r (ξ n ) for ψ 1 (R r ) and m r -a.e. ξ Rr, where W r is given by (). Theorem 3.4. If the conditions (iii) and (iii) in Theorem 3. are replaced by the condition: for each bounded subsetωof C +, there exists a (Borel measurable) function Φ Ω 1 (R r ) satisfying (14) Ψ(λ,η) Φ Ω (η) for all λ Ω and m r -a.e. η Rr, then the conclusions of Theorem 3. hold true.

8 91 DONG HYUN CHO Proof. Suppose that ϕ r is absolutely continuous and Ψ satisfies (i) and (ii) of Theorem 3.. By the same method as used in the proof of Theorem 3. (I λ (F)ψ)(ξ) = λ r (K λ (F))(ξ,ξ 1,...,ξ n )ψ(ξ n )Ψ(λ,ξ ξ) R (n+1)r W r (λ, τ n,(ξ,ξ 1,...,ξ n ))dm r (ξ )dm r (ξ 1 ) dm r (ξ n ) for λ >, ψ 1 (R r ) and ξ R r, where K λ (F) is given by (8). et Ω λ be a bounded subset of C + containing λ. Then for m r -a.e. ξ Rr (I λ (F)ψ)(ξ) λ r MΩλ R (n+1)r ψ(ξ n ) Φ Ωλ (ξ ξ) W r (λ, τ n, (ξ,ξ 1,...,ξ n ))dm r (ξ )dm r (ξ 1 ) dm r (ξ n ) by (14), where Ω is replaced by Ω λ. For j = 1,...,n 1, a simple calculation shows that [ ] r { λ π exp λ ( ξj+1 ξ j R r (t j+1 t j )(t j t j 1 ) R t r j+1 t j + ξ j ξ j 1 )} R r dm r t j t (ξ j) j 1 [ ]r λ = exp { λ ξ j+1 ξ j 1 } R r. π(t j+1 t j 1 ) (t j+1 t j 1 ) Hence ( (I λ (F)ψ)(ξ) λ r λ MΩλ πt { exp λ ξ n ξ R r t )r ψ(ξ n ) Φ Ωλ (ξ ξ) Rr } dm r (ξ,ξ n ) ( ) r λ M Ωλ Φ Ωλ 1(R r ) ψ 1(R r ), πt so that I λ (F)ψ (R r ) and I λ (F) ( 1 (R r ), (R r )). For ψ 1 (R r ) let (Q λ (F)ψ)(ξ) be the right hand side of (1) for (λ,ξ) C + R r. Using the same method as used in the proof of Theorem 3., we can prove the existence of Iλ an (F) and the equality Ian λ (F) = Q λ(f) for λ C + if (1) is replaced by the following inequality (Q λ (F)ψ)(ξ) M Ω λ (n+1)r (Reλ) (n 1)r Φ Ω 1(R r ) ψ 1(R r ) ( 1 πt which is easily obtained for λ Ω and m r -a.e. ξ Rr, where Ω is arbitrary bounded subset of C +. The remainder part of the theorem follows from Theorems 3.5 and 3.7 in [5]. Theorem 3.5. et n, the assumptions be as given in emma. and X n be given by (4). For F : C r [,t] C suppose that [F X n ]( ξ n ) exists for )r

9 OPERATOR-VAUED FUNCTION SPACE INTEGRAS 911 λ C + and m (n+1)r -a.e. ξ n R (n+1)r, and that for each bounded subset Ω of C +, there exists a (Borel measurable) function Ψ Ω 1 (R nr ) such that (15) [F X n ](ξ,ξ 1,...,ξ n 1,ξ n ) Ψ Ω (ξ,ξ 1,...,ξ n 1 ) for all λ Ω and m (n+1)r -a.e. (ξ,ξ 1,...,ξ n ) R (n+1)r. Furthermore suppose that there exists a function Ψ on C + R r satisfying conditions (i), (ii), (iii) of Theorem 3.. Then for λ C +, the analytic operator-valued Wiener wϕ- r integral Iλ an (F) exists as an element of and is given by (1). In addition, suppose that for a nonzero real q, E anfq [F X n ]( ξ n ) exists for m (n+1)r -a.e. ξ n R (n+1)r. Moreover suppose that Ψ can be extended to (C + { iq}) R r with the condition (ii) of Theorem 3.. Then the analytic operator-valued Feynman wϕ r -integral Jan q (F) exists as an element of and it is given by (1) replacing λ and by iq and E anfq, respectively. Proof. Suppose that ϕ r is absolutely continuous and Ψ satisfies (i), (ii) and (iii) of Theorem 3.. By the same method as used in the proof of Theorem 3. (I λ (F)ψ)(ξ) = λ r (K λ (F))(ξ,ξ 1,...,ξ n 1,ξ n )ψ(ξ n )Ψ(λ,ξ ξ) R (n+1)r W r (λ, τ,(ξ,ξ 1,...,ξ n ))dm r (ξ )dm r (ξ 1 ) dm r (ξ n ) for λ >, ψ 1 (R r ) and ξ R r, where K λ (F) is given by (8). et Ω λ be a bounded subset of C + containing λ. Then for m r -a.e. ξ Rr [ n ]r (I λ (F)ψ)(ξ) λ r λ Ψ Ωλ, ψ(ξ n ) π(t j t j 1 ) R(n+1)r Ψ Ωλ (ξ,ξ 1,...,ξ n 1 ) dm r (ξ )dm r (ξ 1) dm r (ξ n) [ n ]r = λ r λ Ψ Ωλ, Ψ Ωλ 1(R nr ) ψ 1(R r ), π(t j t j 1 ) where Ψ Ωλ, denotes the essential supremum of Ψ on Ω λ R r so that I λ (F)ψ (R r ) and I λ (F) ( 1 (R r ), (R r )). For ψ 1 (R r ) let (Q λ (F)ψ)(ξ) be the right hand side of (1) for (λ,ξ) C + R r. Using the same method as used in the proof of Theorem 3., we can prove the existence of Iλ an (F) and the equality Ian λ (F) = Q λ(f) for λ C + if (1) is replaced by the following inequality (Q λ (F)ψ)(ξ) λ r Ψ Ω, Ψ Ω 1(R nr ) ψ 1(R r ) [ n ]r λ π(t j t j 1 ) which is easily obtained for λ Ω and m r -a.e. ξ Rr, where Ω is arbitrary bounded subset of C +. Furthermore suppose that Ψ satisfies (ii) of Theorem 3.. For ψ 1 (R r ) let (Jq an (F)ψ)(ξ) be the right hand side of (1) for ξ R r where λ and are replaced by iq and E anfq, respectively, and

10 91 DONG HYUN CHO let Ω ǫ = {λ C + : λ+iq < ǫ} for some ǫ >. Then, by (ii) of Theorem 3., (J an q (F)ψ)(ξ) q r Ψ Ωǫ, Ψ Ωǫ 1(R nr ) ψ 1(R r ) [ n ( q +ǫ) r Ψ Ωǫ, Ψ Ωǫ 1(R nr ) ψ 1(R r ) ]r q π(t j t j 1 ) [ n ]r q +ǫ π(t j t j 1 ) which implies Jq an(f)ψ (R r ) and Jq an(f) ( 1(R r ), (R r )). The remainder part of the theorem follows from Theorem 3.5 in [5]. By Theorem 3.5 we can obtain the following corollary. Corollary 3.6. et n, X n be given by (4) and ϕ r be the measure as given in Corollary 3.3. Suppose that F : C r [,t] C satisfies the conditions in Theorem 3.5. Then for nonzero real q, the analytic operator-valued Feynman wϕ r -integral Jan q (F) exists as an element of and it is given by the right hand side of (13) replacing λ and by iq and E anfq, respectively. Using the same method as used in the proof of Theorem 3.5, we can prove the following theorem. Theorem 3.7. If the conditions (iii) and (15) in Theorems 3. and 3.5, respectively, are replaced by (14) in Theorem 3.4 and the following condition: for each bounded subset Ω of C +, there exists a (Borel measurable) function Ψ Ω 1 (R (n 1)r ) such that [F X n ](ξ,ξ 1,...,ξ n 1,ξ n ) Ψ Ω (ξ 1,...,ξ n 1 ) for all λ Ω and m (n+1)r -a.e. (ξ,ξ 1,...,ξ n ) R (n+1)r, respectively, then the statement of Theorem 3.5 holds true. Theorem 3.8. et n, the assumptions be as given in emma. and X n be given by (4). For F : C r [,t] C suppose that [F X n ]( ξ n ) satisfies (9) in Theorem 3.. et B(x) = f(x(t ),...,x(t n 1 ))F(x) for wϕ-a.e. r x C r [,t], where f 1 (R nr ). Furthermore suppose that there exists a function Ψ on C + R r satisfying conditions (i), (ii) and (iii) of Theorem 3.. Then for λ C +, the analytic operator-valued Wiener wϕ-integral r Iλ an(b) exists as an element of and is given by (16) λ (B)ψ)(ξ) = λ r f(ξ,ξ 1,...,ξ n 1 ) [F X n ](ξ,ξ 1,...,ξ n 1,ξ n )ψ(ξ n ) R (n+1)r Ψ(λ,ξ ξ)w r (λ, τ,(ξ,ξ 1,...,ξ n ))dm r (ξ )dm r (ξ 1 ) dm r (ξ n ) (I an

11 OPERATOR-VAUED FUNCTION SPACE INTEGRAS 913 for ψ 1 (R r ) and m r -a.e. ξ Rr. In addition, suppose that for a nonzero real q, E anfq [F X n ]( ξ n ) exists for m (n+1)r -a.e. ξn R (n+1)r. Moreover suppose that Ψ can be extended to (C + { iq}) R r with the condition (ii) of Theorem 3.. Then the analytic operator-valued Feynman wϕ-integral r Jq an (B) exists as an element of and it is given by the right hand side of (16), where λ and are replaced by iq and E anfq, respectively. Proof. For λ > and ξ n = (ξ,ξ 1,...,ξ n ) R (n+1)r (K λ (B))( ξ n ) = f(λ 1 (x [x])(t )+[ ξ n ](t ),...,λ 1 (x [x])(tn 1 ) C r +[ ξ n ](t n 1 ))F(λ 1 (x [x])+[ ξn ])dwϕ r (x) = f(ξ,...,ξ n 1 ) F(λ 1 (x [x])+[ ξn ])dwϕ(x) r C r = f(ξ,...,ξ n 1 )(K λ (F))( ξ n ), where K λ is given by (8). Since f(ξ,...,ξ n 1 ) is independent of λ C +, the existence of [B X n ]( ξ n ) follows from the existence of [F X n ]( ξ n ) and [B X n ]( ξ n ) = f(ξ,ξ 1,...,ξ n 1 ) [F X n ]( ξ n ). Furthermore, for any bounded subset Ω of C +, [B X n ]( ξ n ) = f(ξ,ξ 1,...,ξ n 1 ) [F X n ]( ξ n ) M Ω f(ξ,ξ 1,...,ξ n 1 ) for all λ Ω and m (n+1)r -a.e. ξn = (ξ,ξ 1,...,ξ n ) R (n+1)r. The theorem now follows from Theorem 3.5. By Corollary 3.3 and Theorem 3.8, we have the following corollary. Corollary 3.9. et n and X n be given by (4). Moreover let ϕ r be the measure as given in Corollary 3.3. Suppose that B is as given in Theorem 3.8. Then for nonzero real q, the analytic operator-valued Feynman w r ϕ -integral J an q (B) exists as an element of and it is given by (17) = (J an q (B)ψ)(ξ) ( )r q πiα f(ξ,ξ 1,...,ξ n 1 )E anfq [F X n ](ξ,ξ 1,...,ξ n 1,ξ n ) R(n+1)r { qi ξ ξ } R ψ(ξ n )W r ( iq, τ n (ξ,ξ 1,...,ξ n ))exp r α dm r (ξ )dm r (ξ 1) dm r (ξ n ) for ψ 1 (R r ) and m r -a.e. ξ Rr, where W r is given by ().

12 914 DONG HYUN CHO By the same method as used in the proof of Theorem 3.8, we obtain the following theorem from Theorem 3.7. Theorem 3.1. et n, the assumptions be as given in emma. and X n be given by (4). For F : C r [,t] C suppose that [F X n ]( ξ n ) satisfies (9) in Theorem 3.. et D(x) = f(x(t 1 ),...,x(t n 1 ))F(x) for wϕ-a.e. r x C r [,t], where f 1 (R (n 1)r ). Furthermore suppose that there exists a function Ψ on C + R r satisfying conditions (i), (ii) of Theorem 3. and (14) of Theorem 3.4. Then for λ C +, the analytic operator-valued Wiener wϕ r -integral Ian λ (D) exists as an element of and is given by (16), where f(ξ,ξ 1,...,ξ n 1 ) is replaced by f(ξ 1,...,ξ n 1 ). In addition, suppose that for a nonzero real q, E anfq [F X n ]( ξ n ) exists for m (n+1)r -a.e. ξ n R (n+1)r. Moreover suppose that Ψ can be extended to (C + { iq}) R r with the condition (ii) of Theorem 3.. Then the analytic operator-valued Feynman wϕ-integral r (D) exists as an element of and it is given by the right hand side of (16) where λ, and f(ξ,ξ 1,...,ξ n 1 ) are replaced by iq, E anfq and f(ξ 1,...,ξ n 1 ), respectively. J an q Remark The function Ψ satisfying the conditions in Theorems 3., 3.4, 3.5, 3.7, 3.8 and 3.1 exists [5]. We note that such a function which is not a normal density can be obtained. 4. The conditional w ϕ -integrals of bounded functions and the operator-valued function space integrals Throughout this section, we assume that r = 1. et M = M( [,t]) be the class of all C-valued Borel measures of bounded variation over [,t] and let S wϕ be the space of all functions F of the form; for σ M (18) F(x) = exp{i(v, x)}dσ(v) [,t] for w ϕ -a.e. x C[,t]. Using the same method in [3], it can be shown that S wϕ is a Banach algebra. et ˆM(R γ ) be the set of all functions φ on R γ defined by (19) φ(u) = exp{i z,u R γ}dρ(z), R γ for u R γ, where ρ is a complex Borel measure of bounded variation over R γ. 1 For each j = 1,...,n, let α j (s) = χ tj t (tj 1,t j 1 j](s) where s t. et V be the subspace of [,t] generated by {α 1,...,α n } and let V denote the orthogonal complement of V. et P and P be the orthogonal projections from [,t] to V and V, respectively. Then for v [,t] v Pv = P v.

13 OPERATOR-VAUED FUNCTION SPACE INTEGRAS 915 Moreover it is not difficult to show that (v,[x]) = (Pv,x) for x C[,t]. For v [,t] and ξ n = (ξ,ξ 1,...,ξ n ) R n+1, we can show that () (v,[ ξ n n ]) = (Pv)(t j )(ξ j ξ j 1 ). Throughout this paper, let {v 1,v,...,v γ } be an orthonormal subset of [,t] such that {P v 1,..., P v γ } are independent. et {e 1,...,e γ } be the orthonormal set obtained from {P v 1,..., P v γ } by the Gram-Schmidt orthonormalization process. Now, for l = 1,...,γ, let γ (1) P v l = α lj e j be the linear combinations of the e j s and let α 11 α 1 α 1γ α 1 α α γ A = α γ1 α γ α γγ be the coefficient matrix of the combinations. et T A : R γ R γ and T A T : R γ R γ be given by () T A (z) = za and T A T(z) = za T where z is arbitrary row-vector in R γ and A T is the transpose of A. We note that A is invertible so that T A and T A T are isomorphisms. For convenience we introduce useful notations from the Gram-Schmidt orthonormalization process. For v [,t], we obtain an orthonormal set {e 1,..., e γ,e γ+1 } as follows; let { v,ej for j = 1,...,γ (3) c j (v) = v γ l=1 v,e l for j = γ +1 and if c γ+1 (v). Then e γ+1 = [ 1 v c γ+1 (v) γ ] c j (v)e j γ+1 γ+1 (4) v = c j (v)e j and v = [c j (v)].

14 916 DONG HYUN CHO We note that the equalities in (4) hold trivially for the case c γ+1 (v) =. For v [,t] let ( v,x) = ((v 1,x),...,(v γ,x)) for x C[,t]. Theorem 4.1. et X n be given by (4) with r = 1 and G(x) = F(x)φ( v,x) for w ϕ -a.e. x C[,t], where F S wϕ and φ ˆM(R γ ) are given by (18) and (19), respectively. For v [,t] let c j (P v) be given by (3), where v is replaced by P v for j = 1,...,γ. Then for λ C +, [G X n ]( ξ n ) exists for m n+1 -a.e. ξ n R n+1, and it is given by (5) [G X n ]( ξ n ) { = exp i[(v,[ ξ n ])+ z,( v,[ ξ n ]) R γ] 1 [,t] R λ [ T A(z) R γ γ + c(p v),t A (z) R γ + P v ] }dρ(z)dσ(v), where c(p v) = (c 1 (P v),...,c γ (P v)) and T A is given by (). Furthermore, for nonzero real q, E anfq [G X n ]( ξ n ) exists and it is given by the righthand side of (5) where λ is replaced by iq. In particular, if n = 1, then E anfq [G X 1 ](ξ,ξ 1 ) is given by (6) E anfq [G X 1 ](ξ,ξ 1 ) { = exp i ξ [ 1 ξ t γ ] v(s)ds + z j v j (s)ds + 1 [ γ [,t] R t γ qi ( γ ) γ ( z l α lj + v(s)e j (s)ds 1 ) v(s)ds e j (s)ds t l=1 ( γ z l α lj )+ l=1 [v(s)] ds 1 [ ] ]} v(s)ds dρ(z)dσ(v) t for m -a.e. (ξ,ξ 1 ) R, where z = (z 1,...,z γ ) and the α lj s are as given in (1). Proof. The equation (5) and existence of E anfq [G X n ] follow from Theorem.6 in [6]. To prove the remainder part of the theorem, suppose that n = 1 and let ξ 1 = (ξ,ξ 1 ). Then for v [,t] (7) (v,[ ξ 1 ]) = ξ 1 ξ t Furthermore P v = v Pv = v v,α 1 α 1 = v v(s)ds. ( 1 t ) v(s)ds χ (,t]

15 OPERATOR-VAUED FUNCTION SPACE INTEGRAS 917 so that for j = 1,...,γ, (8) c j (P v) = P v,e j = and (9) P v = v v,α 1 α 1 = v(s)e j (s)ds 1 t v(s)ds e j (s)ds [v(s)] ds 1 [ v(s)ds]. t The equation (6) now follows and hence the proof is completed. Corollary 4.. et the assumptions and notations be as given in Theorem 4.1. (1) If σ is concentrated on V, then for nonzero q and m n+1 -a.e. ξn = (ξ,ξ 1,..., ξ n ) R n+1, { [ E anfq [G X n ]( ξ n n ) = exp i v(t j )(ξ j ξ j 1 ) [,t] R γ and E anfq [G X 1 ](ξ,ξ 1 ) = + z,( v,[ ξ n ]) R γ [,t] { [ exp i R γ ] v j (s)ds ] + 1 qi T A(z) R γ + 1 qi v(t)(ξ 1 ξ )+ ξ 1 ξ t } dρ(z)dσ(v) γ z j γ ( γ ) z l α lj }dρ(z)dσ(v). () If σ is concentrated on V, then for nonzero real q, { E anfq [G X n ]( ξ n ) = exp i z,( v,[ ξ n ]) R γ + 1 [,t] R qi [ T A(z) R γ γ } + c(v),t A (z) R γ + v ] dρ(z)dσ(v) and E anfq [G X 1 ](ξ,ξ 1 ) { = exp i ξ 1 ξ γ z j v j (s)ds+ 1 [ γ ( γ ) z l α lj [,t] R t γ qi l=1 γ ( )( γ ]} + v(s)e j (s)ds z l α lj )+ [v(s)] ds dρ(z)dσ(v). l=1 l=1 (3) If v l V for l = 1,...,γ, then for nonzero real q, E anfq [G X n ]( ξ n )

16 918 DONG HYUN CHO { = exp i(v,[ ξ n ])+ 1 } [,t] R qi [ z R γ + c(p v),z R γ + P v ] γ dρ(z)dσ(v), where c(p v) = ( P v,v 1,..., P v,v γ ), and E anfq [G X 1 ](ξ,ξ 1 ) { = exp i ξ 1 ξ [,t] R t γ v j (s)ds 1 t dρ(z)dσ(v). v(s)ds v(s)ds+ 1 [ γ γ ( zj qi + z j v(s) ) v j (s)ds + [v(s)] ds 1 [ ] ]} v(s)ds t Proof. (1) If σ is concentrated on V, then for σ-a.e. v [,t], Pv = v and P v = so that by () (v,[ ξ n n n ]) = (Pv)(t j )(ξ j ξ j 1 ) = v(t j )(ξ j ξ j 1 ) and for j = 1,...,γ c j (P v) = P v,e j =. The results now follow from Theorem 4.1. () If σ is concentrated on V, then for σ-a.e. v [,t], P v = v and Pv = so that (v,[ ξ n ]) = by (). The results now follow from Theorem 4.1. (3) If v l V for l = 1,...,γ, then Pv l = so that by () ( v,[ ξ n ]) = ((v 1,[ ξ n ]),...,(v γ,[ξ n ])) = (,...,) which implies z,( v,[ ξ n ]) R γ =. Furthermore, P v l = v l and e l = v l which implies that A is the identity matrix. By Theorem 4.1, the results follow. Remark 4.3. (1) We note that there exist orthonormal vectors v 1,v,...,v γ in [,t] such that P v 1,P v,...,p v γ are independent [6]. () If v l V for some l, then P v l = and hence P v 1,...,P v γ are dependent. In this case, the proof of Theorem 4.1 can be modified. (3)ettingρ = δ orσ = δ,wecanobtaine anfq [F X n ]ore anfq [φ( v, ) X n ], respectively [6, Theorems.1 and.4]. Since [G X n ]( ξ n ) is bounded by ρ σ, the next theorem follows immediately from Theorems 3. and 4.1. Theorem 4.4. et r = 1, the assumptions be as given in emma., X n be given by (4) and G be as given in Theorem 4.1. Furthermore suppose that there exists a function Ψ on C + R satisfying the conditions (i), (ii) and (iii) in Theorem 3.. Then for λ C +, the analytic operator-valued Wiener w ϕ - integral Iλ an (G) exists as an element of and is given by (1) with r = 1,

17 OPERATOR-VAUED FUNCTION SPACE INTEGRAS 919 where [F X n ] is replaced by [G X n ] which is as given in (5). In addition, suppose that n = 1 and Ψ can be extended to (C + { iq}) R with the conditions (ii) and (iii) of Theorem 3.. Then the analytic operator-valued Feynman w ϕ -integral Jq an (G) exists as an element of and it is given by (11) with r = 1, where E anfq [F X 1 ] is replaced by E anfq [G X 1 ] which is as given in (6). Corollary 4.5. et r = 1, X n be given by (4) and G be as given in Theorem 4.1. Moreover let ϕ be normally distributed with mean and variance α. Then for λ C +, the analytic operator-valued Wiener w ϕ -integral Iλ an (G) exists as an element of and is given by (13) with r = 1, where [F X n ] is replaced by [G X n ] which is as given in (5). Remark 4.6. Under the assumptions as givenin Corollary4.5, we can provethe existence of the analytic operator-valued Feynman w ϕ -integral Jq an (G) through direct calculations, but they are tedious. By Theorems 3.4 and 4.1 we can easily obtain the following theorem. Theorem 4.7. If, in Theorem 4.4, the conditions (iii) and (iii) of Theorem 3. are replaced by (14), then conclusions of Theorem 4.4 hold true. By Theorems 3.8 and 4.1 we can also obtain the following theorem. Theorem 4.8. et r = 1, n, the assumptions be as given in emma., X n be given by (4) and G be as given in Theorem 4.1. et B G (x) = f(x(t ),...,x(t n 1 ))G(x) for w ϕ -a.e. x C[,t], where f 1 (R n ). Furthermore suppose that there exists a function Ψ on C + R satisfying the conditions (i), (ii) and (iii) of Theorem 3.. Then for λ C +, the analytic operator-valued Wiener w ϕ - integral Iλ an(b G) exists as an element of and is given by (16) with r = 1, where [F X n ] is replaced by [G X n ] which is as given in (5). In addition, suppose that for nonzero real q, Ψ can be extended to (C + { iq}) R with the condition (ii) of Theorem 3.. Then the analytic operator-valued Feynman w ϕ -integral Jq an(b G) exists as an element of and it is given by the right hand side of (16) with r = 1, where λ and [F X n ] are replaced by iq and E anfq [G X n ], respectively. Corollary 4.9. et r = 1, n, X n be given by (4) and B G be as given in Theorem 4.8. Moreover let ϕ be normally distributed with mean and variance α. Then for nonzero real q, the analytic operator-valued Feynman w ϕ -integral Jq an (B G ) exists as an element of and it is given by the right hand side of (17) with r = 1, where E anfq [F X n ] is replaced by E anfq [G X n ] which is as given in Theorem 4.1. The following theorem now follows from Theorems 3.1 and 4.1.

18 9 DONG HYUN CHO Theorem 4.1. et r = 1, n, the assumptions be as given in emma., X n be given by (4) and G be as given in Theorem 4.1. et D G (x) = f(x(t 1 ),...,x(t n 1 ))G(x) for w ϕ -a.e. x C[,t], where f 1 (R n 1 ). Furthermore suppose that there exists a function Ψ on C + R satisfying the conditions (i), (ii) of Theorem 3. and (14) of Theorem 3.4. Then for λ C +, the analytic operator-valued Wiener w ϕ -integral Iλ an(d G) exists as an element of and is given by (16) with r = 1, where [F X n ] and f(ξ,ξ 1,...,ξ n 1 ) are replaced by [G X n ] and f(ξ 1,...,ξ n 1 ), respectively. In addition, suppose that for a nonzero real q, Ψ can be extended to (C + { iq}) R with the condition (ii) of Theorem 3.. Then the analytic operator-valued Feynman w ϕ -integral Jq an(d G) exists as an element of and it is given by the expression of Iλ an(d G), where λ is replaced by iq. 5. The conditional w ϕ -integrals of cylinder functions and the operator-valued function space integrals In this section, we investigate the conditional analytic Wiener and Feynman w ϕ -integrals of cylinder functions and prove that the operator-valued function space integrals of those functions can be expressed by the conditional w ϕ - integrals. We now have the following theorem from (), (7), (8), (9) and Theorem 3.3 of [6]. Theorem 5.1. et X n be given by (4) with r = 1 and H(x) = F(x)f( v,x), where f p (R γ )(1 p ) and F is given by (18). For v [,t] let c j (P v) be given by (3), where v is replaced by P v for j = 1,...,γ. Then for λ C +, [H X n ]( ξ n ) exists for ξ n R n+1, and it is given by (3) [H X n ]( ξ n ) ( )γ { λ = f(( v,[ ξ π [,t] n ])+T A T(z))exp i(v,[ ξ n ]) R γ + 1 [ γ ]} [λiz j +c j (P v)] P v dm γ λ (z)dσ(v), where z = (z 1,...,z γ ) and T A T is given by (). In particular, if p = 1, then for nonzero real q, E anfq [H X n ]( ξ n ) exists and it is given by the right hand side of (3) where λ is replaced by iq. Furthermore, if n = 1, then [H X 1 ](ξ,ξ 1 ) is given by (31) [H X 1 ](ξ,ξ 1 )

19 = OPERATOR-VAUED FUNCTION SPACE INTEGRAS 91 ( )γ ( ( λ ξ1 ξ t f π [,t] R t γ γ { α 1j z j,..., α γj z j ))exp i ξ 1 ξ t v(s)e j (s)ds 1 t ]} dm γ (z)dσ(v) v(s)ds v 1 (s)ds,..., e j (s)ds] v(s)ds+ 1 λ for (ξ,ξ 1 ) R, where the α lj s are as given in (1). ) ( γ v γ (s)ds + [ γ [v(s)] ds+ 1 t [ λiz j + [ ] v(s)ds Using the same method as used in the proof of Corollary 4., we can prove the following corollary. Corollary 5.. et the assumptions and notations be as given in Theorem 5.1. (1) If σ is concentrated on V, then for λ C + and ξ n = (ξ,ξ 1,...,ξ n ) R n+1 ( )γ { [H X n ]( ξ λ n ) = f(( v,[ ξ π [,t] n ])+T A T(z))exp i R γ n v(t j )(ξ j ξ j 1 ) λ z R }dm γ γ (z)dσ(v) and, for n = 1 and for (ξ,ξ 1 ) R [H X 1 ](ξ,ξ 1 ) ( )γ ( ( λ ξ1 ξ t ) = f v 1 (s)ds,..., v γ (s)ds π [,t] R t γ ( γ γ + α 1j z j,..., α γj z j ))exp {iv(t)(ξ 1 ξ ) λ } z Rγ dm γ (z)dσ(v) where z = (z 1,...,z γ ). () If σ is concentrated on V, then for λ C + and for ξ n R n+1, ( )γ { [H X n ]( ξ λ n ) = f(( v,[ ξ π [,t] n ])+T A T(z))exp R γ [ γ ]} 1 [λiz j +c j (v)] v dm γ λ (z)dσ(v) where z = (z 1,...,z γ ), and for n = 1 and for (ξ,ξ 1 ) R, [H X 1 ](ξ,ξ 1 )

20 9 DONG HYUN CHO ( )γ ( ( λ ξ1 ξ t ) = f v 1 (s)ds,..., v γ (s)ds π [,t] R t γ ( γ γ { 1 γ [ + α 1j z j,..., α γj z j ))exp λiz j + v(s) λ ]} e j (s)ds] [v(s)] ds dm γ (z 1,...,z γ )dσ(v). (3) If v l V for l = 1,...,γ, then for λ C + and for ξ n R n+1, [H X n ]( ξ n ) ( )γ { λ = f(z 1,...,z γ )exp i(v,[ ξ π n ])+ 1 [ γ [λiz j [,t] R λ γ ]} +c j (P v)] P v dm γ (z 1,...,z γ )dσ(v) and for n = 1 and for (ξ,ξ 1 ) R, = [H X 1 ](ξ,ξ 1 ) ( )γ λ π [,t] [ γ + 1 λ [ λiz j + [v(s)] ds+ 1 t { f(z 1,...,z γ )exp i ξ 1 ξ R t γ [ v(s)ds ] v j (s)ds v(s)v j (s)ds 1 v(s)ds t ] ]} v(s)ds dm γ (z 1,...,z γ )dσ(v). etting σ = δ which is the Dirac measure concentrated at [,t], we obtain the following corollary. Corollary 5.3. et X n be given by (4) with r = 1 and H(x) = f( v,x) where f p (R γ )(1 p ). Then for λ C +, [H X n ]( ξ n ) exists for ξ n R n+1 and it is given by (3) [H X n ]( ξ n ) ( )γ λ = f(( v,[ ξ π n ])+T A T(z))exp { λ } dm γ z Rγ (z), Rγ where T A T is given by (). In particular, if p = 1, then for nonzero real q, E anfq [H X n ]( ξ n ) exists and it is given by the right hand side of (3) where λ is replaced by iq. Furthermore, if n = 1, then [H X 1 ](ξ,ξ 1 ) is given by [H X 1 ](ξ,ξ 1 )

21 OPERATOR-VAUED FUNCTION SPACE INTEGRAS 93 ( )γ ( ( λ ξ1 ξ t ) = f v 1 (s)ds,..., v γ (s)ds π t Rγ ( γ γ + α 1j z j,..., α γj z j ))exp { λ } dm γ z Rγ (z) for (ξ,ξ 1 ) R, where z = (z 1,...,z γ ) and the α lj s are as given in (1). Theorem 5.4. If, in Theorem 4.4, G is replaced by H(p = 1) which is as given in Theorem 5.1, then the conclusions of Theorem 4.4 hold true, where [H X n ] and E anfq [H X 1 ] are given by (3) and (31), respectively, replacing λ by iq. Proof. For λ C +, for ξ n R n+1 and for v [,t], { exp i(v,[ ξ n ])+ 1 [ γ [λiz j +c j (P v)] P v λ ]} { = exp Reλ γ zj Reλ [ γ ]} λ P v P v,e j 1 by (3) and the Bessel s inequality so that ( )γ [H X n ]( ξ λ n ) σ f(( v,[ ξ π n ])+T A T(z)) dm γ (z) Rγ by Theorem 5.1. et Ω be a bounded subset of C + and take M Ω > such that λ M Ω for all λ Ω. Then for λ Ω and for ξ n R n+1, (33) [H X n ]( ξ n ) det((a T ) 1 ) f 1 σ ( MΩ by the change of variable theorem. The theorem now follows from Theorems 3. and 5.1. Corollary 5.5. If, in Corollary 4.5, G is replaced by H which is as given in Theorem 5.4, then the conclusion of the corollary holds true, where [H X n ] is given by (3). By Theorems 3.4, 5.1 and (33) we can easily obtain the following theorem. Theorem 5.6. If, in Theorem 5.4, the conditions (iii) and (iii) of Theorem 3. are replaced by (14), then conclusions of Theorem 5.4 hold true. By Theorems 3.8, 5.1 and (33), we can also obtain the following theorem. Theorem 5.7. If, in Theorem 4.8, G is replaced by H which is as given in Theorem 5.4, then the conclusions of Theorem 4.8 hold true, where [H X n ] and E anfq [H X n ] are as given in Theorem 5.1. π )γ

22 94 DONG HYUN CHO Corollary 5.8. If we replace G in Corollary 4.9 by H which is as given in Theorem 5.4, then the conclusion of the corollary holds true, where E anfq [H X n ] is as given in Theorem 5.1. The following theorem now follows from Theorems 3.1, 5.1 and (33). Theorem 5.9. If, in Theorem 4.1, G is replaced by H which is as given in Theorem 5.4, then the conclusions of Theorem 4.1 hold true, where [H X n ] and E anfq [H X n ] are as given in Theorem 5.1. References [1] R. H. Cameron and D. A. Storvick, An operator-valued function space integral and a related integral equation, J. Math. Mech. 18 (1968), [], An operator-valued function space integral applied to integrals of functions of class 1, Proc. ondon Math. Soc. (3) 7 (1973), [3], Some Banach algebras of analytic Feynman integrable functionals, Analytic functions, Kozubnik 1979 (Proc. Seventh Conf., Kozubnik, 1979), pp , ecture Notes in Math., 798, Springer, Berlin-New York, 198. [4] D. H. Cho, A simple formula for an analogue of conditional Wiener integrals and its applications, Trans. Amer. Math. Soc. 36 (8), no. 7, [5], Operator-valued Feynman integral via conditional Feynman integrals on a function space, Cent. Eur. J. Math. 8 (1), no. 5, [6] D. H. Cho, B. J. Kim, and I. Yoo, Analogues of conditional Wiener integrals and their change of scale transformations on a function space, J. Math. Anal. Appl. 359 (9), no., [7] D. M. Chung, C. Park, and D. Skoug, Operator-valued Feynman integrals via conditional Feynman integrals, Pacific J. Math. 146 (199), no. 1, 1 4. [8] G. B. Folland, Real Analysis, John Wiley & Sons, New York, [9] M. K. Im and K. S. Ryu, An analogue of Wiener measure and its applications, J. Korean Math. Soc. 39 (), no. 5, [1] G. W. Johnson and D.. Skoug, The Cameron-Storvick function space integral: the 1 theory, J. Math. Anal. Appl. 5 (1975), [11] K. S. Ryu and M. K. Im, A measure-valued analogue of Wiener measure and the measure-valued Feynman-Kac formula, Trans. Amer. Math. Soc. 354 (), no. 1, Dong Hyun Cho Department of Mathematics Kyonggi University Suwon , Korea address: j94385@kyonggi.ac.kr