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Unitarily invariant norm inequalities for operators (1101.3895v1)
Published 20 Jan 2011 in math.FA
Abstract: We present several operator and norm inequalities for Hilbert space operators. In particular, we prove that if $A_{1},A_{2},...,A_{n}\in {\mathbb B}({\mathscr H})$, then [|||A_{1}A_{2}{}+A_{2}A_{3}{}+...+A_{n}A_{1}{}|||\leq|||\sum_{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 ${\mathbb B}({\mathscr H})$, then &&|||(\sum_{i=1}{4}(-1){i+1}A_{i})\oplus0\oplus0\oplus0|||&\leq&|||(A_{1}+|A_{3}A_{1}|)\oplus (A_{2}+|A_{4}A_{2}|)\oplus(A_{3}+|A_{1}A_{3}|)\oplus(A_{4}+|A_{2}A_{4}|)||| for any unitarily invariant norm.