Rotfel’d type inequalities for norms

Abstract

In this note, we consider some norm inequalities related to the Rotfel’d Trace Inequality Tr f ( | A + B | ) ⩽ Tr f ( | A | ) + f ( | B | ) for concave functions f : [ 0 , ∞ ) → [ 0 , ∞ ) and arbitrary n-by- n matrices. For instance we show that for a large class of non-negative concave functions f ( t ) and for all symmetric norms we have ‖ f ( | A + B | ) ‖ ⩽ 2 ‖ f ( | A | ) + f ( | B | ) ‖ and we conjecture that this holds for all non-negative concave functions.