New operator inequalities in finite-dimensional vector spaces

Abstract

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