Some Properties of Strictly Positive Doubly Stochastic Matrices

Abstract

The aim of this work is to investigate some properties of the set of all strictly positive doubly stochastic $I\times I$ matrices denoted by $\Omega ^{+}(I),$ which is the subset of all doubly stochastic $I\times I$ matrices denoted by $\Omega (I),$ where I is an arbitrary nonempty set. For uncountable set I,  we have $\Omega ^{+}(I)=\emptyset ,$ so we consider $I=\mathbb {N},$ and in this case $\Omega ^{+}(I)$ and $\Omega (I)$ are denoted by $\Omega ^{+}$ and $\Omega ,$ respectively. We prove that $\textrm{card}\,\Omega ^{+}=\textrm{card}\,\Omega =\textrm{card}\,{\mathbb {R}}$. We show that $\Omega ^{+}$ is closed under countable convex combination. Since $\Omega ^{+}\subset {\mathcal {D}}{\mathcal {S}}(\ell ^{p}),$ where ${\mathcal {D}}{\mathcal {S}}(\ell ^{p})$ is the set of all doubly stochastic operators on $\ell ^{p},$ so we consider p-norm for the elements of $\Omega ^{+}$. Also, for $1\le p<\infty ,$ we show that $\Omega ^{+}$ is not closed. For $1<p<\infty ,$ there exists $D\in \Omega ^{+}$ with $\Vert D\Vert <1$ and $0\in \overline{\Omega ^{+}},$ morever for $\alpha \in (0,1],$ there is $D\in \Omega ^{+}$ such that $\Vert D\Vert =\alpha .$ There exists $D\in \Omega ^{+}$ which is compact. Some relevant examples are indicated.