Title: On the invariant subspace problem in Hilbert spaces

Abstract: Abstract: I will present a method to construct invariant subspaces - non-cyclic vectors - for a general operator on Hilbert space. It represents a new direction of a method of "extremal vectors", first presented in Ansari-Enflo [1]. One looks for an analytic function l(T) of T, of minimal norm, which moves a vector y near to a given vector x. The construction produces for most operators T a non-cyclic vector, by gradual approximation by almost non-cyclic vectors. But for certain weighted shifts, almost non-cyclic vectors will not always converge to a non-cyclic vector. The construction recognizes this, and when the construction does not work, it will show, that T has some shift-like properties. Reference: 1. S. Ansari, P. Enflo, "Extremal vectors and invariant subspaces", Transactions of Am. Math. Soc. Vol. 350 no.2, 1998, pp.539–558