An analog of the Hille theorem for hypercomplex functions in a finite-dimensional commutative algebra

Abstract

We prove that a locally bounded and differentiable in the sense of Gâteaux function given in a finite-dimensional commutative Banach algebra over the complex field is also differentiable in the sense of Lorc and it is a monogenic function. The algebra $\mathbb{A}_n^m$ has the Cartan basis for which the first $m$ basic  vectors $I_1,$ $I_2,$ $\ldots,$ $I_m$ are idempotents, and next $n-m$ basis vectors $I_{m+1},I_{m+2},\dots,I_n$ are nilpotent elements.
Every locally bounded and differentiable in the sense of Gâteaux function $\Phi\colon  \Omega\rightarrow\mathbb{A}_n^m$ can be represented in the form of linear combination of these idempotents, nilpotents and corresponding Cauchy-type integrals.