Unitarily invariant norm inequalities for operators

We present several norm inequalities for Hilbert space operators. In particular, we prove that if A1,A2,…,An∈B(H), then|||A1A2∗+A2A3∗+⋯+AnA1∗|||⩽∑i=1nAiAi∗for all unitarily invariant norms.We also show that if A1,A2,A3,A4A1,A2,A3,A4 are projections in B(H)B(H), then∑i=14(-1)i+1Ai⊕0⊕0⊕0⩽|||(A1+|A3A1|)⊕(A2+|A4A2|)⊕(A3+|A1A3|)⊕(A4+|A2A4|)|||for any unitarily invariant norm.