International Seminar Series on Advances in Operator Theory and Inequalities #2

We study the approximate minimizing property (AMp) for operators, a localized Bishop-Phelps-Bollob\'{a}s type property with respect to minimum norm. Given Banach spaces $X$ and $Y$ we define a new class $\mathcal{AM}(X,Y)$ of bounded linear operators from $X$ to $Y$ for which the pair $(X,Y)$ satisfies the AMp. We provide a necessary and sufficient condition for non-injective operators from $X$ to $Y$ to be in the class $\mathcal{AM}(X,Y)$. We also prove that $X$ is finite dimensional if and only if for every Banach space $Y$, $(X,Y)$ has the AMp for all minimum norm attaining operators from $X$ to $Y$ if and only if for every Banach space $Y$, $(Y,X)$ has the AMp for all minimum norm attaining operators from $Y$ to $X$. We also study the AMp with respect to Crawford number called AMp-$c$ for operators.

This paper is devoted to refining several results on reverse inequalities for the numerical radius and the operator norm of operators on Hilbert spaces.

A tournament is $k$-spectrally monomorphic if all the $k\times k$ principal submatrices of its adjacency matrix have the same characteristic polynomial. Transitive $n$-tournaments are trivially $k$-spectrally monomorphic. We show that there are no others for $k\in\{3,\ldots,n-3\}$. Furthermore, we prove that for $n\geq 5$, a non-transitive $n$-tournament is $(n-2)$-spectrally monomorphic if and only if it is doubly regular. Finally, we give some results on $(n-1)$-spectrally monomorphic regular tournaments.