Related papers: Generalized Bernstein operators defined by increas…

The cut and join operations play important roles in tensor models in general. We introduce a generalization of the cut operation associated with the higher order variations and demonstrate how they generate operators in the Aristotelian…

This paper presents an algebraic approach to characterizing higher-order differential operators. While the foundational Leibniz rule addresses first-order derivatives, its extension to higher orders typically involves identities relating…

Let $F$ and $G$ be two bounded operators on two Hilbert spaces. Let their numerical radii be no greater than one. This note investigate when there is a $\Gamma$-contraction $(S,P)$ such that $F$ is the fundamental operator of $(S,P)$ and…

In this article we present the Durrmeyer variant of generalized Bernstein operators that preserve the constant functions involving non-negative parameter ?. We derive the approximation behaviour of these operators including global…

To extend the Euclidean operator radius, we define $w_p$ for an $n$-tuples of operators $(T_1,\ldots, T_n)$ in $\mathbb{B}(\mathscr{H})$ by $w_p(T_1,\ldots,T_n):= \sup_{\| x \| =1} \left(\sum_{i=1}^{n}| \langle T_i x, x \rangle |^p…

The $\alpha$-Bernstein operators were initially introduced in the paper by Chen, X., Tan, J., Liu, Z., Xie, J. (2017) titled "Approximation of Functions by a New Family of Generalized Bernstein Operators" (Journal of Mathematical Analysis…

Let $ \mathcal{B}:=\{f(z)=\sum_{n=0}^{\infty}a_nz^n\; \mbox{with}\; |f(z)|<1\;\mbox{for all}\; z\in\mathbb{D}\} $. The improved version of the classical Bohr's inequality \cite{Bohr-1914} states that if $ f\in\mathcal{B} $, then the…

We study in this paper properties of functions of perturbed normal operators and develop earlier results obtained in \cite{APPS2}. We study operator Lipschitz and commutator Lipschitz functions on closed subsets of the plane. For such…

The effect of modifying General Relativity with the addition of some higher dimensional operators, generalizations of the Goroff-Sagnotti operator, is discussed. We determine in particular, the general solution of the classical equations of…

It is well known that a Lipschitz function on the real line does not have to be operator Lipschitz. We show that the situation changes dramatically if we pass to H\"older classes. Namely, we prove that if $f$ belongs to the H\"older class…

We give several new characterizations of completely monotone functions and Bernstein functions via two approaches: the first one is driven algebraically via elementary preserving mappings and the second one is developed in terms of the…

A common theme in mathematics is to define generalized solutions to deal with problems that potentially do not have solutions. A classical example is the introduction of least squares solutions via the normal equations associated with a…

We give a general expression for the normally ordered form of a function F(w(a,a*)) where w is a function of boson annihilation and creation operators satisfying [a,a*]=1. The expectation value of this expression in a coherent state becomes…

In this paper we define generalizations of boson normal ordering. These are based on the number of contractions whose vertices are next to each other in the linear representation of the boson operator function. Our main motivation is to…

Generalized Zeckendorf decompositions are expansions of integers as sums of elements of solutions to recurrence relations. The simplest cases are base-$b$ expansions, and the standard Zeckendorf decomposition uses the Fibonacci sequence.…

In this short note, we establish the following result: Let $f:[0,+\infty[\to [0,+\infty[$, $\alpha:[0,1]\to ]0,+\infty[$ be two continuous functions, with $f(0)=0$. Assume that, for some $a>0$, the function $\xi\to…

Let $A,$ $T$ and $B$ be bounded linear operators on a Banach space. This paper is concerned mainly with finding some necessary and sufficient conditions for convergence in operator norm of the sequences $\left\{ A^{n}TB^{n}\right\} $ and…

The purpose of this survey is a comprehensive study of operator Lip\-schitz functions. A continuous function $f$ on the real line ${\Bbb R}$ os called operator Lipschitz if $\|f(A)-f(B)\|\le\operatorname{const}\|A-B\|$ for arbitrary…

Let $A$ and $B$ be two -non necessarily bounded- normal operators. We give new conditions making their product normal. We also generalize a result by Deutsch et al on normal products of matrices.

We show how series expansions of functions of bosonic number operators are naturally derived from finite-difference calculus. The scheme employs Newton series rather than Taylor series known from differential calculus, and also works in…