New operator inequalities in finite-dimensional vector spaces

Abstract:

We establish some new operator inequalities in an $n$-dimensional vector space $X$ equipped with a seminorm $\|\cdot \|$. Here is an example. If $A$ is an invertible linear operator in $X$ and $\xi$ is a vector, then \[ \|\xi \|^r \le \sum _{1\le |j|\le {n+r-1\choose r}}\|A^j\xi \|^r.\] Some special cases have been known and used in mathematical physics.