Geodesic
On Square Roots and Logarithms of Self-Adjoint Operators
Glasgow mathematical journal, Tome 4 (1958) no. 1, pp. 1-2

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All operators considered in this paper are bounded and linear (everywhere defined) on a Hilbert space. An operator A will be called a square root of an operator B ifA simple sufficient condition guaranteeing that any solution A of (1) be normal whenever B is normal was obtained in [1], namely: If B is normal and if there exists some real angle θ for which Re(Aeιθ)≥0, then (1) implies that A is normal. Here, Re (C) denotes the real part 1⁄2(C + C*) of an operator C. 
Putnam, C. R. On Square Roots and Logarithms of Self-Adjoint Operators. Glasgow mathematical journal, Tome 4 (1958) no. 1, pp. 1-2. doi: 10.1017/S204061850003375X
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[1] 1.Putnam, C. R., On square roots of normal operators, Proc. Amer. Math. Soc., 8 (1957), 768–769. Google Scholar | DOI

[2] 2.Wecken, F. J., Zur Theorie linearer Operatoren, Math. Annalen 110 (1935), 722–725. Google Scholar

[3] 3.Wintner, A., Ueber das Aequivalenzproblem beachrankter hermitescher Formen, Math. Z., 37 (1933), 254–263. Google Scholar | DOI

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