A generalized operational calculus developed from Fredholm operator theory

Abstract:

Let A be a closed operator on the Banach space X. We construct an operator, ${R’_\lambda }(A)$, depending on the parameter, $\lambda$, and having the following properties: \[ \begin {array}{*{20}{c}} {(\lambda - A){{R’}_\lambda }(A) = I + {F_1},} \\ {{{R’}_\lambda }(A)(\lambda - A) = I + {F_2}} \\ \end {array} \] where ${F_1}$ and ${F_2}$ are bounded finite rank operators. ${R’_\lambda }(A)$ is defined and analytic in $\lambda$ for all $\lambda \in {\Phi _A}$ except for at most a countable set containing no accumulation point in ${\Phi _A}$. Let ${\sigma _\Phi }(A)$ be the complement of ${\Phi _A}$, and let $f \in {\mathcal {A}’_\infty }(A)$, where ${\mathcal {A}’_\infty }(A)$ denotes the set of complex valued functions which are analytic on ${\sigma _\Phi }(A)$ and at $(\infty )$. We then use the operator, ${R’_\lambda }(A)$, to construct an operational calculus for A. $f(A)$ is defined up to addition by a compact operator. We prove for our operational calculus analogues of the theorems for the classical operational calculus. We then extend a theorem of Kato by using the operator, ${R’_\lambda }(A)$, to construct an analytic basis for $N(A - \lambda )$.