Appl Math Optim 48:67–92 (2003)
2003 Springer-Verlag New York Inc.
A Generalized Analytic Operator-Valued Function Space
Integral and a Related Integral Equation
Kun Soo Chang,
Byoung Soo Kim,
Cheong Hee Park,
and Kun Sik Ryu
Department of Mathematics, Yonsei University,
Seoul, 120-749, Korea
Department of Computer Science, University of Minnesota,
Minneapolis, MN 55455, USA
Department of Mathematics, Hannam University,
Daejon, 330-791, Korea
Communicated by G. Kallianpur
Abstract. We introduce a generalized Wiener measure associated with a Gaussian
Markov process and deﬁne a generalized analytic operator-valued function space
integral as a bounded linear operator from L
(1 < p ≤ 2) by the an-
alytic continuation of the generalized Wiener integral. We prove the existence of
the integral for certain functionals which involve some Borel measures. Also we
show that the generalized analytic operator-valued function space integral satisﬁes
an integral equation related to the generalized Schr¨odinger equation. The resulting
theorems extend the theory of operator-valued function space integrals substantially
and previous theorems about these integrals are generalized by our results.
Key Words. A generalized Wiener measure space, A Gaussian Markov process,
A generalized analytic operator-valued function space integral.
AMS Classiﬁcation. Primary 28C20, Secondary 60G15.
This research was supported by the Basic Science Research Institute Program, Korea Research Foun-
dation under Grant KRF 2001-005-D20004.