Kernel Summability Methods and Spaces Of Holomorphic Functions

We introduce kernel summability methods in Banach spaces. We then extend the Silverman-Toeplitz Theorem to these summability methods. We also show that if $X$ is a Banach space and one kernel summability methods is included in another kernel summability method for scalar-valued functions, then the first method is included for $X$-valued functions in the second method. This extends a previous result from Javad Mashreghi, Thomas Ransford and the author. We then give some general applications on the summability of Taylor series of functions in a Banach space of holomorphic functions on the unit disk

Counting Involutions on Multicomplex Numbers

We show that there is a bijection between real-linear automorphisms of the multicomplex numbers of order $n$ and signed permutations of length $2^{n-1}$. This allows us to deduce a number of results on the multicomplex numbers, including a formula for the number of involutions on multicomplex spaces which generalizes a recent result on the bicomplex numbers and contrasts drastically with the quaternion case. We also generalize this formula to $r$-involutions and obtain a formula for the number of involutions preserving elementary imaginary units. The proofs rely on new elementary results pertaining to multicomplex numbers that are surprisingly unknown in the literature, including a count and a representation theorem for numbers squaring to $\pm 1$