A remark concerning a theorem of B. Friedman
J. Feldman · Proceedings of the American Mathematical Society · 1958
In [1], the following theorem is stated: let T be a densely defined linear operator with closed range in the Hilbert space X, with a densely defined adjoint T* also having closed range. Let X, i1 be vectors in 3C, and let '0'k1 be the operator defined by f4 1V(x) = (x, 0)41. Then T+0 0'1 also has closed range. Of course, the fact that T* is densely defined implies that T is pre-closed; but an examination of the proof shows that it actually requires that T be a closed operator. Under this assumption, a simpler proof can be given; and the need for some such condition will be shown by example.