Title: On the Invariant Subspace Problem in Hilbert Space

Abstract:

A method to construct invariant subspaces for a general operator T on Hilbert space is presented. It represents a new direction of my "method of extremal vectors", first presented in 1998 in Ansari-Enflo [1].

The Main Construction of the method gives a non-cyclic vector of T by gradual approximation by "almost non-cyclic" vectors. There are reasons, why the Main Construction cannot work for some weighted shifts. But when the Main Construction fails, one gets, by using the information obtained, invariant subspaces of T, similar to those of weighted shifts.

The sequence y(n) of "almost non-cyclic" vectors follows the formula y(n+1) = y(n) + r(T) y(n), which will allow for efficient use of elementary Fourier Analysis. The more general formula y(n+1) = y(n) + z would lead to difficult problems concerning the relation between T and T*, problems which may be of interest in themselves.

Reference:

1. S.Ansari, P.Enflo, "Extremal vectors and invariant subspaces", Transactions of Am. Math. Soc. Vol. 350 no.2, 1998, pp.539-558