Finite-Rank Perturbations of Normal Operators: Hyperinvariant Subspaces and a Problem of Pearcy

Finite-rank perturbations of diagonalizable normal operators acting boundedly on infinite-dimensional, separable, complex Hilbert spaces are considered from the standpoint of view of the existence of invariant subspaces. In particular, if T = D-Lambda + u circle times v is a rank-one perturbation of a diagonalizable normal operator D-Lambda with respect to a basis E = {e(n)}(n >= 1 )and the vectors u and v have Fourier coefficients {alpha(n)}(n >= 1 )and {beta(n)}(n >= 1 )with respect to E, respectively, it is shown that T has non-trivial closed invariant subspaces provided that either u or v have a Fourier coefficient which is zero or u and v have non-zero Fourier coefficients and & sum;(n >= 1)|alpha(n)|(2)log1/|alpha n| + |beta(n)|(2 )log1/|beta n| < infinity. As a consequence, if (p, q) is an element of (0,2] x (0,2] are such that & sum;(infinity)(n=1)(|alpha(n)|p + |beta(n)|q) < infinity, the existence of non-trivial closed invariant subspaces of T is shown when-ever (p, q) is an element of (0,2] x (0,2] \ {(2, r ), (r ,2) : r is an element of (1,2]}. Moreover, such operators T have non-trivial closed hyperinvariant sub-spaces whenever they are not a scalar multiple of the identity. Likewise, analogous results hold for finite-rank perturbations of D-Lambda. This improves considerably previous theorems of Foias, Jung, Ko, and Pearcy [5], Fang and Xia [4], and the authors [8] on an open question explicitly posed by Pearcy in the seventies.