Inequalities for trace on $tau$-measurable operators

Let $mathfrak{M}$ be a semifinite von Neumann algebra on a Hilbert space equipped with a faithful normal semifinite trace $tau$. A closed densely defined operator $x$ affiliated with $mathfrak{M}$ is called $tau$-measurable if there exists a number $lambda geq 0$ such that $tau left(e^{|x|}(lambda,infty)right)<infty$. A number of useful inequalities, which are known for the trace on Hilbert space operators, are extended to trace on $tau$-measurable operators. In particular, these inequalities imply Clarkson inequalities for $n$-tuples of $tau$-measurable operators. A general parallelogram law for $tau$-measurable operators are given as well.

Bohrs inequality revisited

We survey several significant results on the Bohr inequality and presented its generalizations in some new approaches. These are some Bohr type inequalities of Hilbert space operators related to the matrix order and the Jensen inequality. An eigenvalue extension of Bohrs inequality is discussed as well.

Eigenvalue extensions of Bohrs inequality

We present a weak majorization inequality and apply it to prove eigenvalue and unitarily invariant norm extensions of a version of the Bohrs inequality due to Vasic and Kev{c}kic.

Dunkl--Williams inequality for operators associated with $p$-angular distance

We present several operat