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We give two new sufficient conditions for unbounded Hilbert space operators to be subnormal. The first assumes that the sequence //Tnf//2 on a suitable subset of the domain is completely monotonic, the second is similar to the one given by Lambert in  for bounded operators and involves the sequence of binomial expansion of the real part of the operator.doi:10.1017/s0017089501010035 fatcat:ntzzu6tuxze3bbisih3rmopqqa