Sufficient Conditions for the Invariant Subspace Problem

In this note, we provide a few sufficient conditions for the invariant subspace problem. Introduction 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. Thus, if S = T −λIH is one-to-one, then we always assume that S has a dense range. We provide sufficient conditions for the invariant subspace problem in Theorem 2.6 ; (a) If S = T − λIH is not one-to-one, then T has a non-trivial invariant subspace. (b) If S is one-to-one and there is λ ∈ σ(T ) such that limn→∞ S ∗ 1 (xn) = 0 for any sequence {xn} ∞ n=0 in D(S ∗ 1 ) satisfying limn→∞ xn = 0, then T has a non-trivial invariant subspace. 1. Preliminaries and Notation In this note, C and L(H) denote the set of complex numbers, and the set of bounded linear operators from H to H where H is a separable Hilbert space whose dimension is not finite, respectively. For any vectors h1 and h2 in H , (h1, h2) denotes the inner product of h1 and h2.