• 대한수학회지
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• 제31권3호
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• pp.375-391
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• 1994

In 1968k Cameron and Storvick introduced the analytic and the sequential operator-valued function space integral [2]. Since then, the theo교 of the analytic operator-valued function space integral has been investigated by many mathematicians - Cameron, Storvick, Johnson, Skoug, Lapidus, Chang and author etc. But there are not that many papers related to the theory of the sequential operator-valued function space integral. In this paper, we establish the existence of the sequential operator-valued function space integral as an operator from $L_p$ to $L_p'(1 and investigated the integral equation related to this integral.

• 대한수학회논문집
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• 제9권3호
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• pp.599-606
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• 1994

In [1], Cameron and Storvick introduced the analytic operator-valued function space integral. Johnson and Lapidus proved that this integral can be expressed in terms of an integral of operator-valued functions [6]. In this paper, we find some operator-valued Bochner integrable functions and prove that the analytic operator-valued function space integral of a certain function is represented as the Bochner integral of operator-valued functions on some conditions.

• 대한수학회보
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• 제53권3호
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• pp.903-924
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• 2016

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 operator-valued 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.

• 충청수학회지
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• 제23권1호
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• pp.37-48
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• 2010

In this paper we use a generalized Brownian motion process to defined an analytic operator-valued generalized Feynman integral. We then obtain explicit formulas for the analytic operatorvalued generalized Feynman integrals for functionals of the form $$F(x)=f\({\int}^T_0{\alpha}_1(t)dx(t),{\cdots},{\int}_0^T{\alpha}_n(t)dx(t)\)$$, where x is a continuous function on [0, T] and {${\alpha}_1,{\cdots},{\alpha}_n$} is an orthonormal set of functions from ($L^2_{a,b}[0,T]$, ${\parallel}{\cdot}{\parallel}_{a,b}$).

• 호남수학학술지
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• 제29권1호
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• pp.9-18
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• 2007

In this paper we establish the existence theorem of the operator-valued function space integral $K_{\lambda}(F)$ for $\lambda\in\mathbb{C}_{+,\lambda_0}^\sim$ where F is of the form (1.1). Our result is a generalization for Johnson and Skoug result in [8].

• 한국수학사학회지
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• 제14권2호
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• pp.21-28
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• 2001

In this paper we introduce the Feynman integral which is one of the function space integrals. There are so many approaches to the Feynman integral. Here we treat tile analytic Feynman integral and the operator-valued Feynman integral.

• 대한수학회보
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• 제26권1호
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• pp.11-17
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• 1989

In [1], Johnson and Lapidus introduced a family { $A_{t}$ :t>0} of Banach algebras of functionals on Wiener space and showed that for every F in $A_{t}$ , the analytic operator-valued function space integral $K_{\lambda}$$^{t}$ (F) exists for all nonzero complex numbers .lambda. with nonnegative real part. In [2,3] Johnson and Lapidus introduced a noncommtative multiplication having the property that if F.mem. $A_{t}$ $_{1}$ and G.mem. $A_{t}$ $_{2}$ then $F^{*}$G.mem. A$t_{1}$+$_{t}$ $_{2}$ and (Fig.) Note that for F, G in $A_{t}$ , $F^{*}$G is not in $A_{t}$ but rather is in $A_{2t}$ and so the multiplication * is not internal to the Banach algebra $A_{t}$ . In this paper we introduce an internal noncommutative multiplication on $A_{t}$ having the property that for F, G in $A_{t}$ , F G is in $A_{t}$ and (Fig.) for all nonzero .lambda. with nonnegative real part. Thus is an auxiliary binary operator on $A_{t}$ .TEX> .