Consider a separable infinite dimensional Hilbert space H with an orthonormal basis (x_n). Let D = {h Î H ;å_{n = 1}^¥ n⁴| < h,x_n > |² < ¥} , z = å_{n = 2}^¥ n^-1x_n, and define the operators S and T with the domain D, which is dense in H, by

Sh = ¥ å n = 2  n2 < h,xn > xn    ,    Th = Sh+ < Sh,z > x1      (h Î D).

Then -S and T are closed densely defined and T + (-S) isn't closable. (cf. Problem 2.8.43 of [K&R1])
Ref
[K&R1] R.V. Kadilon and J.R. Ringrase, Fundamentals of the theory of operator algebras (I), Acad. Press, 1983.