Refinements of operator Cauchy-Schwarz and Minkowski inequalities for p-modified norms and related norm inequalities

Abstract

For (degree) p-modifications Φ(p) of symmetrically norming functions Φ and finite families of bounded Hilbert space operators A1,B1,⋯,AN, BN and X∑n=1NAn∗XBNΦ(p)≤||(|∑n=1NAn∗X Bn2 + ∑n=1N|An(||∑n=1NAn∗An||1/2I+(||∑n=1NAn∗An||I-∑n=1NAn∗An)1/2)-1 ∑m=1NAm∗XBm-||∑n=1NAn∗An||1/2XBN|2)p/2||Φ1p=||∑n=1NAn∗An||1/2||(∑n=1NBn∗X∗XBN)1/2||Φ(p) for 0<p≤2, while for p≥2||∑n=1NAn∗XBn||Φ(p)≤|||∑n=1NAn∗XBn|p + ∑n=1N |An (||∑n=1NAn∗An||1/2I+(||∑n=1NAn∗An|| I-∑n=1NAn∗An)1/2)-1∑m=1NAm∗XBm-||∑n=1NAn∗An||1/2XBN|p||Φ1/p≤||∑n=1NAn∗An||1/2||∑n=1NBn∗X∗XBn||Φ(p2). For p≥2 this implies refined Minkowski inequality for p-modified unitarily invariant norms ||·||Φ(p) ||∑n=1NBN||Φ(p)≤|||∑n=1N Bn|p +(∑n=1N||Bn||Φ(p))-p/2∑n=1N||Bn||Φ(p)-p/2|||Bn||Φ(p)∑m=1NBm-∑n=1m||Bm||Φ(p)BN|p||Φ1/p≤∑n=1n||Bn||Φ(p).