A Solution to the Invariant Subspace Problem

An important open problem in operator theory is the invariant subspace problem. Since the problem is solved for all finite dimensional complex vector spaces of dimension at least 2, H denotes a separable Hilbert space whose dimension is infinite. It is enough to think for a contraction T , that is, ‖T‖ ≤ 1 on H . Thus, T ( 6= 0) denotes a contraction in L(H). Let λ always denote an element of the spectrum σ(T ) of T . If λ ∈ σ(T ) such that S = T − λIH is one-to-one, and has dense range, then S1(: Range(S) → H) always denotes a densely defined operator defined by S1(a) = S (a) for a in the range of S, where S(a) denotes an element such that S(S(a)) = a. Clearly, S1 is a linear mapping, and S1 is a densely defined operator in H . It is not assumed that S1 is bounded or continuous. If S = T−λIH is one-to-one, then clearly S is not onto. If S( 6= 0) does not have a dense range, then the closure of Range (S) is a non-trivial invariant subspace of S and so is of T . Thus, if S = T − λIH is one-to-one, then we always assume that S has a dense range. In the proof of Proposition 2.3, we show that S 1 is also densely defined. We provide sufficient conditions for the invariant subspace problem in Theorem 2.6 ;