Normal operators and multipliers on complex Banach spaces and a symmetry property of \(L^ 1\)-predual spaces (Q796804)

An operator T on a complex Banach space X is called normal if there exists an operator S such that \(({1\over2})(T+S)\) and \(({1\over2}i)(T-S)\) are Hermitian and \(TS=ST.\) We show how \(T:X\to X\) is normal if and only if \(T':X'\to X'\) is normal. Using a generalization of the principle of local reflexivity this result enables us to prove that multipliers on complex \(L^ 1\)-predual spaces are always normal.