We study the operator monotonicity of the inverse of every polynomial with a positive leading coefficient. Let {pn}n=0 be a sequence of orthonormal polynomials and pn+ the restriction of pn to [an,∞), where an is the maximum zero of pn. Then p −1 n+ and the composite pn−1 ◦ p −1 n+ are operator monotone on [0,∞). Furthermore, for every polynomial p with a positive leading coefficient there is a...

• Notation. What will here be called density operators are often (as in CQT) referred to as density matrices. Alas, a “density matrix” is not always a matrix, and has nothing whatsoever to do with “density” in the usual meaning of that term. Thus the term “density operator,” which only suffers from the second of these defects, is to be preferred. ⋆ Mathematical definition: any operator ρ on the...

We construct monotone numerical schemes for a class of nonlinear PDE for elliptic and initial value problems for parabolic problems. The elliptic part is closely connected to a linear elliptic operator, which we discretize by monotone schemes, and solve the nonlinear problem by iteration. We assume that the elliptic differential operator is in the divergence form, with measurable coefficients s...

Abstract In this paper, we focus on a $2 \times 2$ 2 × operator matrix $T_{\epsilon _{k}}$ T ϵ k as follows: $$\begin{aligned} T_{\epsilon _{k}}= \begin{pmatrix} A & C \\ \epsilon _{k} D B\end{pmatrix}, \end{aligned...