Completely Indecomposable Operators and a Uniqueness Theorem of Cartwright–Levinson Type

Abstract

A bounded linear operator T on a complex Hilbert space will be called completely indecomposable if its spectrum is not a singleton, and is included in the spectrum of the restrictions of T and T * to any of their nonzero invariant subspaces. Two classes of completely indecomposable operators are constructed. The first consists of essentially selfadjoint operators with spectrum [&2, 2], and the second of bilateral weighted shifts whose spectrum is the unit circle. We do not know whether any of the operators in the first class has a proper invariant subspace and if any of the operators in the second class has a proper hyperinvariant subspace. We also establish a new uniqueness theorem of Cartwright Levinson type which is the main ingredient in our proofs of complete indecomposability. 1999Academic Press

On the local spectral properties of weighted shift operators

“…Hence, (ω n ) n∈Z is in particular almost periodic and r 1 (S) = r(S) = 1. On the other hand, it is shown in [6] that (ω n ) n∈Z satisfies all the hypotheses of Theorem 2 of [6]. It follows from the proof of that theorem that σ S (x) = σ S * (x) = σ(S) = {λ ∈ C : |λ| = 1} for every non-zero x ∈ H. This shows that both S and S * have fat local spectra, and so satisfy Dunford's condition (C), but neither S nor S * has Bishop's property (β).…”

Reducible representations of abelian groups

“…[1][2][3][4][5][6][7][8], [12][13][14][15][16][17][18][19][20], [10], [24]). Some comments on the problem for the group R (the discrete real line) appear in [14].…”

Similarity and the Point Spectrum of Some Non-Selfadjoint Jacobi Matrices

“…It seems that there are not many papers devoted to spectral analysis of non-selfadjoint Jacobi matrices, but see [1], [2], [3], [4], [5] and [11]. Note that in the case α n = β n , Jacobi operators have a close relation to the theory of formal orthogonal polynomials (in particular in the study of their asymptotics) and continued fractions.…”