Recent developments of Schwarz’s type trace inequalities for operators in Hilbert spaces. (English) Zbl 1353.47030

Dragomir, Silvestru Sever

Adv. Oper. Theory 1, No. 1, 15-91 (2016).

Summary: In this paper, we survey some recent trace inequalities for operators in Hilbert spaces that are connected to Schwarz’s, Buzano’s and Kato’s inequalities and the reverses of Schwarz inequality known in the literature as Cassels’ inequality and Shisha-Mond’s inequality. Applications for some functionals that are naturally associated to some of these inequalities and for functions of operators defined by power series are given. Examples for fundamental functions such as the power, logarithmic, resolvent and exponential functions are provided as well.

MSC:

47A63

Linear operator inequalities

Keywords:

trace class operators; Hilbert-Schmidt operators; trace; Schwarz inequality; Kato inequality; Cassels’ inequality; Shisha-Mond inequality; trace inequalities for matrices; power series of operators