Daniil Homza | Searcharxiv

arXiv2024

The algebraic numerical range as a spectral set in Banach algebras

We investigate when the algebraic numerical range is a $C$-spectral set in a Banach algebra. While providing several counterexamples based on classical ideas as well as combinatorial Banach spaces, we discuss positive results for matrix algebras and provide an absolute constant in the case of complex $2\times2$-matrices with the induced $1$-norm. Furthermore, we discuss positive results for infinite-dimensional Banach algebras, including the Calkin algebra.

math.FA↗

arXiv2022

Counting Unions of Schreier Sets

A subset of positive integers $F$ is a Schreier set if it is non-empty and $|F|\leqslant \min F$ (here $|F|$ is the cardinality of $F$). For each positive integer $k$, we define $k\mathcal{S}$ as the collection of all the unions of at most $k$ Schreier sets. Also, for each positive integer $n$, let $(k\mathcal{S})^n$ be the collection of all sets in $k\mathcal{S}$ with the maximum element equal to $n$. It is well-known that the sequence $(|(1\mathcal{S})^n|)_{n=1}^\infty$ is the Fibbonacci sequence. In particular, the sequence satisfies a linear recurrence. We generalize this statement, namely, we show that the sequence $(|(k\mathcal{S})^n|)_{n=1}^\infty$ satisfies a linear recurrence for every positive $k$.

math.CO↗