(a) An injective operator on a Hilbert space H such that the range of T, R(T), isn’t dense in H.

(b) An operator S such that S is surjective but Ker(S) ≠ {0}.

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Let H be a separable Hilbert space with the standard orthonormal basis (e_{n}).

(a) The unilateral shift operator T (α_{1}, α_{2}, ⋯) = (0, α_{1}, α_{2}, ⋯) on H is injective and the closure of its range is the closed linear span {e_{2}, e_{3}, ⋯} which doesn’t contain e_{1}.

(b) If S = T^{∗}, then S(α_{1}, α_{2}, ⋯) = (α_{2}, α_{3}, ⋯). So S is surjective but Ker(S) ≠ {0}, since it is the linear span of e_{1}.