In this article we study the Heinz and Hermite-Hadamard inequalities. We derive whole series of refinements these inequalities involving unitarily invariant norms, which improve some recent results, known from literature. We also prove that if $A , B, X\in M_n(\mathbb{C})$ such $A$ $B$ are positive definite $f$ is an operator monotone function on $(0,\infty)$. Then \begin{equation*} |||f(A)X-...

The Haagerup norm ‖ · ‖h on the tensor product A ⊗ B of two C∗-algebras A and B is shown to be Banach space equivalent to either the Banach space projective norm ‖ · ‖γ or the operator space projective norm ‖ · ‖∧ if and only if either A or B is finite dimensional or A and B are infinite dimensional and subhomogeneous. The Banach space projective norm and the operator space projective norm are ...

The first main result in this article provides an error bound for balanced truncation where the matrix norm used is a general Schatten norm rather than the usual operator norm. The second main result in this article is that for the Schatten 1-norm (the trace class norm) this bound, for systems with a semi-definite Hankel operator, is in fact an equality. This class of systems for which we obtai...