Unitarily invariant norm inequalities for operators

Abstract

We present several norm inequalities for Hilbert space operators. In particular, we prove that if A 1 , A 2 , … , A n ∈ B ( H ) , then | | | A 1 A 2 ∗ + A 2 A 3 ∗ + ⋯ + A n A 1 ∗ | | | ⩽ ∑ i = 1 n A i A i ∗ for all unitarily invariant norms. We also show that if A 1 , A 2 , A 3 , A 4 are projections in B ( H ) , then ∑ i = 1 4 ( - 1 ) i + 1 A i ⊕ 0 ⊕ 0 ⊕ 0 ⩽ | | | ( A 1 + | A 3 A 1 | ) ⊕ ( A 2 + | A 4 A 2 | ) ⊕ ( A 3 + | A 1 A 3 | ) ⊕ ( A 4 + | A 2 A 4 | ) | | | for any unitarily invariant norm.