Related papers: A Riesz-Thorin type interpolation theorem in Eucli…
In a Euclidean Jordan algebra V of rank n which carries the trace inner product, to each element x we associate the eigenvalue vector whose components are the eigenvalues of x written in the decreasing order. For any number p between (and…
Given a linear map $T$ on a Euclidean Jordan algebra of rank $n$, we consider the set of all nonnegative vectors $q$ in $R^n$ with decreasing components that satisfy the pointwise weak-majorization inequality…
In an Euclidean Jordan algebra V of rank n, an element x is said to be majorized by an element y, if the corresponding eigenvalue vector of x is majorized by the eigenvalue vector of y in R^n. In this article, we describe pointwise…
In this article, the authors introduce a class of mixed-norm Herz spaces, $\dot{E}^{\vec{\alpha},\vec{p}}_{\vec{q}}(\mathbb{R}^{n})$, which is a natural generalization of mixed Lebesgue spaces and some special cases of which naturally…
Let $\tau$ be a strongly $(n,p;a,c)$ regular graph,such that $0<c<p<n-1,$ $A$ his matrix of adjacency and let ${\cal V}_{n}$ be the Euclidean space spanned by the powers of $A$ over the reals where the scallar product $\bullet|\bullet$ is…
We reconsider the theory of Lagrange interpolation polynomials with multiple interpolation points and apply it to linear algebra. For instance, $A$ be a linear operator satisfying a degree $n$ polynomial equation $P(A)=0$. One can see that…
We investigate the metric structure of nonassociative $\mathrm{L}^p$-spaces associated with tracial $\mathrm{JW}^*$-algebras. While noncommutative $\mathrm{L}^p$-spaces arising from von Neumann algebras enjoy a unique natural norm, the…
In a case study on asymptotics of spectral quantities of Schr\"odinger operators we show how the Riesz-Thorin theorem on the interpolation of linear operators can be extended to nonlinear maps.
This expository thesis contains a study of four interpolation theorems, the requisite background material, and a few applications. The materials introduced in the first three sections of Chapter 1 are used to motivate and prove the…
We consider a two parameter family of Drinfeld twists generated from a simple Jordanian twist further twisted by 1-cochains. Twists from this family interpolate between two simple Jordanian twists. Relations between them are constructed and…
New simple proofs are given to some elementary approximate and explicit inversion formulas for Riesz potentials. The results are applied to reconstruction of functions from their integrals over Euclidean planes in integral geometry.
Let $M_n$ be the algebra of $n \times n$ complex matrices. We consider arbitrary subalgebras $\mathcal{A}$ of $M_n$ which contain the algebra of all upper-triangular matrices (i.e.\ block upper-triangular subalgebras), and their Jordan…
Let V be a Euclidean Jordan algebra of rank n. The eigenvalue map from V to R^n takes any element x in V to the vector of eigenvalues of x written in the decreasing order. A spectral set in V is the inverse image of a permutation set in R^n…
A unified approach to the determination of eigenvalues and eigenvectors of specific matrices associated with directed graphs is presented. Matrices studied include the distance matrix, distance Laplacian, and distance signless Laplacian, in…
For a linear transformation A from Rn to Rn, we give sharp bounds for the average distortion of A, that is, the average value of log of the euclidean norm of Au over all unit vectors u. This is closely related to the results of the author's…
Corresponding to a hyperbolic system $(V, p, e)$, where $V$ is a real finite-dimensional vector space and $p$ is a hyperbolic polynomial of degree $n$ in the direction $e$, we consider the eigenvalue map $\lambda: V \to R^n$ and the…
This work provides a complete characterization of the solutions of a linear interpolation problem for vector polynomials. The interpolation problem consists in finding n scalar polynomials such that an equation involving a linear…
We announce here a number of results concerning representation theory of the algebra $R=k<x,y>/ (xy-yx-y^2)$, known as Jordan plane (or Jordan algebra). We consider the question on 'classification' of finite-dimensional modules over the…
By the Riesz representation theorem using the Riemann-Stieltjes integral, linear continuous functionals on the set of continuous functions from the unit interval into the reals can either be characterized by functions of bounded variation…
It is known that any normed vector space which satisfies the parallelogram law is actually an inner product space. For finite dimensional normed vector spaces over R, we formulate an approximate version of this theorem: if a space…