Related papers: Linear operators on polynomials preserving roots i…

In this note we study the structure of shift-preserving operators acting on a finitely generated shift-invariant space. We define a new notion of diagonalization for these operators, which we call s-diagonalization. We give necessary and…

Let $\A$ be an algebra and let $f(x_1,...,x_d)$ be a multilinear polynomial in noncommuting indeterminates $x_i$. We consider the problem of describing linear maps $\phi:\A\to \A$ that preserve zeros of $f$. Under certain technical…

Based on both the fundamental theorem of affine geometry in regular $L^0$-modules and the recent progress in random convex analysis, this paper characterizes the stable fully order preserving and order reversing operators acting on the…

It is well known that increasing functions do not preserve operator order in general; nor do decreasing functions reverse operator order. However, operator monotone increasing or operator monotone decreasing do. In this article, we employ a…

We compare some properties of the lowering and raising operators for the classical and free classes of Meixner polynomials on the real line.

This is a straightforward introduction to the properties of polynomials in many variables that do not vanish in the open upper half plane. Such polynomials generalize many of the well-known properties of polynomials with all real roots.

In this work, we solve the fundamental problem of describing the coordinate transformations that preserve the upper triangular Toeplitz form of the given operator field. Surprisingly, this problem is closely related to the description of…

Let $H$ be a separable complex Hilbert space. Denote by ${\mathcal G}_{\infty}(H)$ the Grassmannian consisting of closed linear subspaces with infinite dimension and codimension. This Grassmannian is partially ordered by the inclusion…

Let $D\subset \mathbb{R}^d$ be a bounded Lipschitz domain, $\omega$ be a high order modulus of continuity and let $T$ be a convolution Calder\'{o}n-Zygmund operator. We characterize the bounded restricted operators $T_D$ on the Zygmund…

In their classic 1914 paper, Polya and Schur introduced and characterized two types of linear operators acting diagonally on the monomial basis of R[x], sending real-rooted polynomials (resp. polynomials with all nonzero roots of the same…

We completely describe all finite difference operators of the form $$ \Delta_{M_1, M_2, h}(f)(z)=M_1(z) f(z+h) + M_2(z) f(z-h) $$ preserving the Laguerre-P\'olya class of entire functions. Here $M_1$ and $M_2$ are some complex functions and…

In this paper, we are concerned with conditions under which $[p(T)]^*=\bar{p}(T^*)$, where $p(z)$ is a one-variable complex polynomial, and $T$ is an unbounded, densely defined, and linear operator. Then, we deal with the validity of the…

Many interesting families of polynomials are indexed by permutations or related objects, and are defined by applying divided difference operators, modified by polynomials, on some initial base case. The fact that these constructions produce…

We characterize bijections on matrix spaces (operator algebras) preserving full rank (invertibility) of differences of matrix (operator) pairs in both directions.

Representations of polynomial covariance commutation relations by pairs of linear integral and differential operators are constructed in the space of infinitely continuously differentiable functions. Representations of polynomial covariance…

We prove that the diagonal operator defined by a positive sequence preserves tropical and central indices if and only if the sequence is log-concave. In particular we obtain an elementary proof of that such an operator preserves the set of…

We start with a random polynomial $P^{N}(z)$ of degree $N$ with independent coefficients. We then consider a new polynomial $P_{t}^{N}$ obtained by $\lceil Nt\rceil$ applications of a fractional differential operator of the form $z^{a}…

Separable Frobenius monoidal functors were defined and studied under that name by Szlachanyi and by Day and Pastro, and in a more general context by Cockett and Seely. Our purpose here is to develop their theory in a very precise sense. We…

We develop general expressions for the raising and lowering operators that belong to the orthogonal polynomials of hypergeometric type with discrete and continuous variable. We construct the creation and annihilation operators that…

The matrix units of a digraph algebra, A, induce a relation, known as the diagonal order, on the projections in a masa in the algebra. Normalizing partial isometries in A act on these projections by conjugation; they are said to be order…