Operator-valued Feynman integrals of certain finite-dimensional functionals

Let ${C_0}[a,b]$ denote the space of continuous functions $x$ on $[a,b]$ such that $x(a) = 0$. Let $F(x) = {f_1}(x({t_1})) \cdots {f_n}(x({t_n}))$ where $a = {t_0} < {t_1} < \cdots < {t_n} = b$. Recently, Cameron and Storvick defined an operator-valued “Feynman Integral.” In their setting, we give a strong existence theorem as well as an explicit formula for the “Feynman Integral” of functionals $F$ as above under weak restrictions on the ${f_i}$’s. We also give necessary and sufficient conditions for the operator to be invertible and an explicit formula for the inverse.