We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
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 
$2\times 2$-matrices with the induced 
$1$-norm. Furthermore, we discuss positive results for infinite-dimensional Banach algebras, including the Calkin algebra.
A subset of positive integers F is a Schreier set if it is nonempty 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 maximum element equal to n. It is well known that the sequence 
$(|(1\mathcal {S})^n|)_{n=1}^\infty $ is the Fibonacci sequence. In particular, the sequence satisfies a linear recurrence. We show that the sequence 
$(|(k\mathcal {S})^n|)_{n=1}^\infty $ satisfies a linear recurrence for every positive k.
