Product integrals and exponentials in commutative Banach algebras

Abstract:

Functions are from $R \times R$ to $X$, where $R$ represents the real numbers and $X$ represents a commutative Banach algebra with identity element. The function $G \in O{C^ \circ }$ on $[a,b]$ only if ${}_a{\prod ^b}(1 + G)$ exists and is not zero and there exists a subdivision $D$ of $[a,b]$ and a number $B$ such that if $J$ is a refinement of $D$, then ${[\prod \nolimits _J {(1 + G)} ]^{ - 1}}$ exists and $|{[\prod \nolimits _J {(1 + G)} ]^{ - 1}}| < B$. If $|G| < 1$ on $[a,b]$, then each of the following consists of two equivalent statements: A. (1) $G \in O{C^ \circ }$ on $[a,b]$, and (2) $\int _a^b {\ln (1 + G)}$ exists. B. (1) $G \in O{C^ \circ }$ on $[a,b]$ and $\int _a^b {|1 + G - \prod {(1 + G)} | = 0}$, and (2) $\int _a^b {|\ln (1 + G) - \int {\ln (1 + G)|} = 0}$. Further, if $\beta > 0,|G| < 1 - \beta$ on $[a,b]$, each of $G(p,{p^ + }),G({p^ - },p),G({p^ + },{p^ + })$ and $G({p^ - },{p^ - })$ exist for $p \in [a,b],\int _a^b {|{G^2} - \int {{G^2}|} } = 0$ and ${G^2}$ has bounded variation on $[a,b]$, then each of the following consists of two equivalent statements: C. (1) $G \in O{C^ \circ }$ on $[a,b]$, and (2) $\int _a^b G$ exists. D. (1) $G \in 0{C^ \circ }$ on $[a,b]$ and $\int _a^b {|1 + G} - \prod {(1 + G)} | = 0$, and (2) $\int _a^b {|G - \int {G|} = 0}$.