As a cornerstone of functional analysis, Hahn-Banach theorem constitutes an indispensable tool of modern analysis where its impact extends beyond the frontiers of linear functional analysis into several other domains of mathematics, including complex analysis, partial differential equations and ergodic theory besides many more. The paper is an attempt to draw attention to certain applications of the Hahn-Banach theorem which are less familiar to the mathematical community, apart from highlighting certain aspects of the Hahn-Banach phenomena which have spurred intense research activity over the past few years, especially involving operator analogues and nonlinear variants of this theorem. For a discussion of a whole lot of issues related to the Hahn-Banach theorem not treated in this paper, the best source is a famous survey paper by Narici and Beckenstein [31] which deals, among other things, with the different settings witnessing the validity of the Hahn-Banach theorem.

Assume that the linear matrix equation $AXA^{*} = B = B^{*}$ has a Hermitian solution and is partitioned as $\left[\!\! \begin{array}{c} A_1 \\ A_2 \end{array} \!\!\right]\! X[\, A^{*}_1, \, A^{*}_2 \,] = \left[\!\! \begin{array}{cc} B_{11} \,, B_{12} \\ B_{21}^{*} \,, B_{22} \end{array}\right]$. We study in this paper relations among the Hermitian solutions of the equation and the three small-size matrix equations $A_1X_1A^{*}_1 = B_{11}$, $A_1X_2A^{*}_2 = B_{12}$ and $A_2X_3A^{*}_2 = B_{22}$. In particular, we establish closed-form formulas for calculating the maximal and minimal ranks and inertias of $X-X_1-X_2-X_2^{*}-X_3$, and use the formulas to derive necessary and sufficient conditions for the Hermitian matrix equality $X = X_1+X_2+X_2^{*}+X_3$ to hold and Hermitian matrix inequalities $X \geqslant(\leqslant, \cdots)\, X_1+X_2+X_2^{*}+X_3$ to hold in the Löwner partial ordering.

For a subset $E$ of the bidisc $D^2, M=\{f\in H^2(D^2)~:~f=0$ on $E\}$ and $N$ is the orthogonal complement of $M$ in $H^2(D^2)$ where $H^2(D^2)$ is the two variable Hardy space on $D^2$. We describe the finite rank commutants of the restricted shifts $S_z$ and $S_w$ on $N$ when $E$ satisfies some natural condition. Moreover we give a sufficient condition for that the Pick interpolation is possible.

Exponential monomials are the basic building bricks of spectral analysis and spectral synthesis on Abelian groups. Recently there have been some attempts to extend the most important spectral analysis and spectral synthesis results from groups to hypergroups. For this purpose it is necessary to introduce a reasonable concept of exponential monomials. In this work we reconsider this problem, and using a ring-theoretical approach we prove characterization theorems for particular function classes, which can be considered as "exponential monomials" on commutative hypergroups.

In this paper some new Hardy-type inequalities on time scales are derived and proved using the concept of superquadratic functions. Also, we extend Hardy-type inequalities involving superquadratic functions with general kernels to the case with arbitrary time scales. Several consequences of our results are given and their connection with recent results in the literature are pointed out and discussed.

We prove that the non-commutative perspective of an operator convex function is the unique extension of the corresponding commutative perspective that preserves homogeneity and convexity.

Let $H:M_m\to M_m$ be a holomorphic function of the algebra $M_m$ of complex $m\times m$ matrices. Suppose that $H$ is orthogonally additive and orthogonally multiplicative on self-adjoint elements. We show that either the range of $H$ consists of zero trace elements, or there is a scalar sequence $\{\lambda_n\}$ and an invertible $S$ in $M_m$ such that $$ H(x) =\sum_{n\geq 1} \lambda_n S^{-1}x^nS, \quad\forall x \in M_m,%\eqno{(\ddag)}$$ or $$ H(x) =\sum_{n\geq 1} \lambda_n S^{-1}(x^t)^nS, \quad\forall x \in M_m. $$ Here, $x^t$ is the transpose of the matrix $x$. In the latter case, we always have the first representation form when $H$ also preserves zero products. We also discuss the cases where the domain and the range carry different dimensions.

In this paper, we study a class of representations of arithmetic functions, and corresponding operator-theoretic and free probabilistic properties. We associate given arithmetic functions $f$ to certain matrices $\alpha _{n}(f).$

There has been considerable work on matrix approximation problems in the space of matrices with Euclidean and unitarily invariant norms. We initiate the study of approximation problems in the space $\mathbb{P}$ of all $n\times n$ positive definite matrices with the Riemannian metric $\delta_2$. Our main theorem reduces the approximation problem in $\mathbb{P}$ to an approximation problem in the space of Hermitian matrices and then to that in $\mathbb{R}^n$. We find best approximants to positive definite matrices from special subsets of $\mathbb{P}$. The corresponding question in Finsler spaces is also addressed.

In this paper, we discuss exiting roots of sub-kernel transient matrices $P$ associated with a class of $M-$ matrices which are related to generalized ultrametric matrices. Then the results are used to describe completely all links of the class of matrices in terms of structure of the supporting tree.

Given three $n$-tuples $\{\lambda_i\}_{i=1}^n,\{\mu_i\}_{i=1}^n,\{\nu_i\}_{i=1}^n$ of complex numbers, we introduce the problem of when there exists a pair of normal matrices $A$ and $B$ such that $\sigma(A)=\{\lambda_i\}_{i=1}^n, \sigma(B)=\{\mu_i\}_{i=1}^n,$ and $\sigma(A+B)=\{\nu_i\}_{i=1}^n,$ where $\sigma(\cdot)$ denote the spectrum. In the case when $\lambda_k=0,k=2,\ldots,n,$ we provide necessary and sufficient conditions for the existence of $A$ and $B$. In addition, we show that the solution pair $(A,B)$ is unique up to unitary similarity. The necessary and sufficient conditions reduce to the classical A. Horn inequalities when the $n$-tuples are real.

In this paper, by using Mond-Pe\v cari\' c method we provide some inequalities for connections of positive definite matrices. Next, we discuss specifications of the obtained results for some special cases. In doing so, we use $\alpha$-arithmetic, $\alpha$-geometric and $\alpha$-harmonic operator means.

The conic structure of the convex cone of non-negative operator convex functions on $(0,\infty)$ (also on $(-1,1)$) is clarified. We completely determine the extreme rays, the closed faces, and the simplicial closed faces of this convex cone.

We extend Jordan's notion of principal angles to work for two subspaces of quaternionic space, and so have a method to analyze two orthogonal projections in the matrices over the real, complex or quaternionic field (or skew field). From this we derive an algorithm to turn almost commuting projections into commuting projections that minimizes the sum of the displacements of the two projections. We quickly prove what we need using the universal real $C^{*}$-algebra generated by two projections.

In celebration of the distinguished achievements of Professor Tsuyoshi Ando in matrix analysis and operator theory, we conducted an interview with him via email. This paper presents Professor Ando's responses to several questions we gave him regarding his education and life as a mathematician.