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...

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...

In this paper, by employing the positivity of certain block operator matrices, we establish a new class of mixed Schwarz-type inequalities. Our results provide a unified framework that not only extends but also refines several classical inequalities in operator theory. In particular, the obtained inequalities encompass and generalize well-known results such as the mixed Schwarz inequality of...

In this talk, generalized Cauchy-Schwarz inequalities for positive sesquilinear maps with values in noncommutative
Lp-spaces for p > 1 are obtained. Bound estimates for their real and
imaginary parts are also provided, and, as an application, a generalization of the uncertainty relation in the context of noncommutative L2-spaces is given. Next, a Cauchy-Schwarz inequality
for positive...

The talk is based on Dirichlet-type spaces on the unit bidisc. We shall begin with a notion of Dirichlet-type spaces $\mathcal{D}(\mu_1, \mu_2)$ on the unit bidisc $\mathbb{D}^2$ with harmonic weights corresponding to finite positive Borel measures $\mu_1$ and $\mu_2$ supported on the unit circle. Then we show that the coordinate functions $z_1$ and $z_2$ are multipliers for...

This work aims to introduce a new Buzano-type inequality that integrates and unifies several well-established results from the literature. As a consequence, we present novel numerical radius bounds for operators in semi-Hilbertian spaces. For example, it is proven that
for $T \in \mathcal{L}_{A}(\mathcal{H})$ and a mapping $\chi: [0,1]\subset J \rightarrow...

Inspired by the renowned characterization of isometries by Blanco, Koldobsky and Turnšek, this study focuses on the approximate preservation of Birkhoff-James orthogonality by linear operators on normed linear spaces. Specifically, we explore geometric and analytic aspects of this preservation within finite-dimensional polyhedral Banach spaces. The findings of this work yield refined versions...

Let $U,V\subset \mathbb R^{n }$ be two domains and $(s,t) \subset \mathbb R$ be an open interval such that there exists a $C^1$ diffeomorphism $\underline{a} \in C^1( U, V)$. Define $a=I\times \underline{a} : (s,t)\times U \to (s,t)\times V$ given by $y= a(x)$ where $x_0 =(a(x))_0 = y_0$ and $\underline{y}= \underline{ a}( \underline{ x})$. In addition, denote $a(x) =x_0 + ...