Factoring a Quadratic Operator as a Product of Two Positive Contractions

Let T be a quadratic operator on a complex Hilbert space H. We show that T can be written as a product of two positive contractions if and only if T is of the form aI circle plus bI circle plus [ GRAPHICS ] on H-1 circle plus H-2 circle plus (H-3 circle plus H-3) for some a, b is an element of [ 0, 1 ] and strictly positive operator P with parallel to P parallel to \u3c = vertical bar root a - root b vertical bar root (1 - a) (1 - b). Also, we give a necessary condition for a bounded linear operator T with operator matrix [GRAPHICS] on H circle plus K that can be written as a product of two positive contractions

Nonsurjective zero product preservers between matrices over an arbitrary field

In this paper, we give concrete descriptions of additive or linear disjointness preservers between matrix algebras over an arbitrary field $\mathbb{F}$ of different sizes. In particular, we show that a linear map $\Phi: M_n(\mathbb{F}) \rightarrow M_r(\mathbb{F})$ preserving zero products carries the form $$\Phi(A)= S\begin{pmatrix} R\otimes A & 0 \cr 0 & \Phi_0(A)\end{pmatrix} S^{-1},$$ for some invertible matrices $R$ in $M_k(\mathbb{F})$, $S$ in $M_r(\mathbb{F})$ and a zero product preserving linear map $\Phi_0: M_n(\mathbb{F}) \rightarrow M_{r-nk}(\mathbb{F})$ with range consisting of nilpotent matrices. Here, either $R$ or $\Phi_0$ can be vacuous. The structure of $\Phi_0$ could be quite arbitrary. We classify $\Phi_0$ with some additional assumption. When $\Phi(I_n)$ has a zero nilpotent part, especially when $\Phi(I_n)$ is diagonalizable, we have $\Phi_0(X)\Phi_0(Y) = 0$ for all $X, Y$ in $M_n(\mathbb{F})$, and we give more information about $\Phi_0$ in this case. Similar results for double zero product preservers and orthogonality preservers are obtained.Comment: 29 page

Agricultural Sector Input Technical Coefficients, Demand Changes and CO2 Emissions after the Financial Crisis: Environmental Input-Output Growth Factor Model Approach

The agricultural sector has been declining year by year with the proportion of economic growth and GDP. The financial crisis in 2007 caused huge losses in the world's economies. Taiwan cannot avoid economic damage. In the future, the way from agriculture to production needs to be transformed. This study uses the Environmental Input-Output Growth Factor model to estimate the changes in CO2 emissions in the agricultural sector before and after the financial crisis, and summarizes the changing factors to observe the development characteristics of the agricultural sector. The results show that there are differences in the influencing factors before and after the financial crisis. The biggest influencing factors are “domestic final demand” and “production input technical coefficients”. Keywords: Agricultural Sector, CO2 Emission, Input Technical Coefficients, Environmental Input-Output Growth Factor model JEL Classifications: Q15, C6, Q5 DOI: https://doi.org/10.32479/ijeep.702