Clarkson–McCarthy Inequalities for Several Operators and Related Norm Inequalities for p-Modified Unitarily Invariant Norms

Abstract

Let | | · | | Φ be a unitarily invariant norm related to a symmetrically norming (s.n.) function Φ , defined on the associated ideal C Φ (H) of compact Hilbert space operators, let ||·||Φ(q) be its degree q-modification, let ||·||Φ(q)∗ be a dual norm to ||·||Φ(q) and let [Am,n]m,n∈Z be a block operator matrix. We show that, if 0 < p≤ 2 and q≥ p, then ∥[Am,n]m,n∈Z∥Φ(q)p≤∑m∈Z∥[Am,n]n∈Z∥Φ(q)p≤∑m,n∈Z∥Am,n∥Φ(q)p.If 2 ≤ p< + ∞ and q≥ p/ (p- 1) , then ∥[Am,n]m,n∈Z∥Φ(q)∗p≥∑m∈Z∥[Am,n]n∈Z∥Φ(q)∗p≥∑m,n∈Z∥Am,n∥Φ(q)∗p.If 2 ≤ p< + ∞, q≥ p/ (p- 1) and Φ(q)∗=Ψ(r), for some 1 ≤ r≤ p and for some s.n. function Ψ , we extend Clarkson–McCarthy inequalities to an n-tuple of operators (A1,A2,⋯,AN) as N∑n=1N∥An∥Φ(q)∗p≤(∑n=1N∥∑k=1NωNnkAk∥Φ(q)∗r)pr≤Npr-1∑n=1N∥∑k=1NωNnkAk∥Φ(q)∗p≤Npr+p-2(∑n=1N∥An∥Φ(q)∗r)pr≤N2pr+p-3∑n=1N∥An∥Φ(q)∗p.In addition, we provide some refinements of the above inequalities, as well as some new norm inequalities.