I’m very new to this forum, I apologize in advance if I make any mistakes of not abiding by the forum rules.
I am trying to prove the regularity of Borel summation method when we have a series of matrices instead of scalars.
Borrowing from Wikipedia, let $A(z) = \sum_{k=0}^\infty a_kz^k$ be a formal power series. Borel’s integral summation method can be written as
$\int_{0}^\infty e^{-t}\mathcal{B}A(tz)dt,$
where $\mathcal{B}A(tz) = \sum_{n=0}^\infty \frac{t^n}{n!}A_n(z)$ denotes the Borel transform of A and $A_n(z)$ is the partial sum up to the $n$th index. We know that whenever $A(z)$ converges, then the integral also converges to the same value.
I want to think of the situation where $Z \in \mathbb{R}^{n \times n}$, in which case the matrix Borel transform might be written as $\mathcal{B}A(T) = \sum_{k=0}^\infty \frac{a_k}{k!}T^k,$ a function of matrices, and the integral as $\int_{\mathbb{R}_{+}^{n \times n}}e^{-T}\mathcal{B}A(TZ)dT$, and hence I want to show
$\sum_{k=0}^\infty a_kZ^k = Z \implies \int_{\mathbb{R}_{+}^{n \times n}}e^{-T}\mathcal{B}A(TZ)dT = Z,$ under the right conditions.
My question is, what would be the most sensible way to define this integral? My thoughts:
With respect to the eigenvalues of T, i.e. $\int_{\mathbb{R}_+^{n\times n}} (\cdot) dT := \int_0^\infty \int_0^\infty \cdots \int_0^\infty (\cdot) d\lambda_1 d\lambda_2 \cdots d\lambda_k$ where $\lambda_1, \ldots, \lambda_k$ are the eigenvalues of $T$
With respect to every entry: $\int_{\mathbb{R}_+^{n\times n}} (\cdot) dT := \int_0^\infty \int_0^\infty \cdots \int_0^\infty (\cdot)dT_{11}\cdots dT_{nn}.$
I’d appreciate any insight or help!