Geodesic
Normal operators on Banach spaces
Glasgow mathematical journal, Tome 20 (1979) no. 2, pp. 163-168

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A (bounded, linear) operator H on a Banach space is said to be hermitian if ∥exp(itH)∥ = 1 for all real t. An operator N on is said to be normal if N = H + iK, where H and K are commuting hermitian operators. These definitions generalize those familiar concepts of operators on Hilbert spaces. Also, the normal derivations defined in [1] are normal operators. For more details about hermitian operators and normal operators on general Banach spaces, see [4]. The main result concerning normal operators in the present paper is the following theorem. 
Fong, Che-Kao. Normal operators on Banach spaces. Glasgow mathematical journal, Tome 20 (1979) no. 2, pp. 163-168. doi: 10.1017/S0017089500003888
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