The Cauchy-Schwarz Norm Inequality for Elementary Operators in Schatten Ideals

Cauchy–Schwarz norm inequalities for weak∗-integrals of operator valued functions

“…By virtue of (a2) case of Lemma 3.1, the lower semi-continuity of the norm will secure the passage to the general case. & Now, we present an improvement of our Lemma 3.1(c), which also improves (discrete) Theorem 2.1 of [19] and enhances it to the non-discrete case as well. jjA Ã t f jj 2 þ jjB t f jj 2 þ jjB Ã t f jj 2 dmðtÞoN for all f AH; then…”

“…We will try to present some quite new examples, as well as some standard ones but in a new light, so the potential of those inequalities becomes more apparent. The first one will be the Aczel-Bellman inequality for operator integrals; it complements basic Theorem 3.2 and generalizes Theorem 2.3 of [18] in two directions. …”

“…It is now easy to get a non-discrete version of Theorem 2.2 of Jocic´ [18], which will be the one mostly applied in this paper. according to the monotonicity of (singular values and) u.i.…”

“…It is the arithmetic-geometric-logarithmic (56),(74) and Young (66) inequalities that have been under especially intensive investigation and a very good account can be found in [2,6,16]. Power and other various natural mean operator inequalities are investigated in [18,20,21,16], perturbation and generalized derivations norm inequalities in [1,11,17], and different Cauchy-Schwarz inequalities in [9,18,19], with the numerous references therein.…”

“…The goal of this work is to fully develop discrete results of [19] to show the practical potential of the Cauchy-Schwarz inequalities based on the Gel'fand's integration for the more systematic and unifying approach to the above mentioned, as well as some new inequalities and related problems.…”