On two problems concerning linear transformations in hilbert space

T n "0 We shall denote by Cf and C$ the closed linear manifolds spanned by { f}o and {T*'g}o, respectively; f , g being elements in H. This study is devoted to two general problems concerning the transformations T and T* which we shall call the extinction problem and the closure problem. We shall say that T has an extinction theorem if, for every f # o, it is true that the manifold Cj contains at least one eigenelement 9 # o. In the ease