Hyponormality of finite rank perturbations of normal operators

k-HYPONORMALITY OF FINITE RANK PERTURBATIONS OF UNILATERAL WEIGHTED SHIFTS

Abstract. In this paper we explore finite rank perturbations of unilateral weighted shifts Wα. First, we prove that the subnormality of Wα is never stable under nonzero finite rank pertrubations unless the perturbation occurs at the zeroth weight. Second, we establish that 2-hyponormality implies positive quadratic hyponormality, in the sense that the Maclaurin coefficients ofDn(s) := detPn [(W...

Invariant Subspaces for Certain Finite-rank Perturbations of Diagonal Operators

Suppose that {ek} is an orthonormal basis for a separable, infinite-dimensional Hilbert space H. Let D be a diagonal operator with respect to the orthonormal basis {ek}. That is, D = ∑∞ k=1 λkek⊗ek, where {λk} is a bounded sequence of complex numbers. Let T = D + u1 ⊗ v1 + · · ·+ un ⊗ vn. Improving a result [2] of Foias et al., we show that if the vectors u1, . . . , un and v1, . . . , vn satis...

Rank one perturbations of self-adjoint operators

Hyponormality of Trigonometric Toeplitz Operators

In this paper we establish a tractable and explicit criterion for the hyponormality of arbitrary trigonometric Toeplitz operators, i.e., Toeplitz operators Tφ with trigonometric polynomial symbols φ. Our criterion involves the zeros of an analytic polynomial f induced by the Fourier coefficients of φ. Moreover the rank of the selfcommutator of Tφ is computed from the number of zeros of f in the...

On domains of powers of linear operators and finite rank perturbations

Let S and T be linear operators in a linear space such that S ⊂ T . In this note an estimate for the codimension of domSn in domT n in terms of the codimension of domS in domT is obtained. An immediate consequence is that for any polynomial p the operator p(S) is a finite-dimensional restriction of the operator p(T ) whenever S is a finite-dimensional restriction of T . The general results are ...