Dichotomy, discrete Bohl exponents, and spectrum of block weighted shifts

Spectrum of bilateral shifts with operator-valued weights

“…He then deduced that the spectrum of a hyponormal bilateral operator-weighted shift of multiplicity k (i.e., H = C k ) is the union of at most k annuli (see [5,Corollary 3]). In [2], Ben-Artzi and Gohberg introduced the concepts of Bohl exponent and canonical splitting projection to describe the spectrum and the essential spectrum of S when H = C k and max sup…”

Fine Structure of the Dichotomy Spectrum

“…We point out that such a bridge was probably first observed by Mather [34] (see also the monographs [14,21]). Related ideas were later taken up by [5,7], who characterize the spectrum σ(T A ) for difference eqns. (Δ) defined on the whole integer axis Z, as the union of up to d concentric annuli, where d is the finite dimension of the state space for (Δ).…”

“…(Δ) defined on the whole integer axis Z, as the union of up to d concentric annuli, where d is the finite dimension of the state space for (Δ). The radii of these annuli are so-called Bohl exponents and [7] obtain explicit forms of the spectrum σ(T A ) for constant, periodic, asymptotically constant and scalar sequences A k . An identical description of the set σ(T A ) using local spectral theory can be found in [ the single value extension property (SVEP) is given in [11].…”

“…[49] resp. [3,7])-equivalently one also speaks of the dynamical or Sacker-Sell spectrum. In stability theory the dichotomy spectrum extends the role of the classical spectrum σ(A) from an autonomous setting x k+1 = Ax k to a general time-dependent set-up of general eqns.…”

The single-valued extension property for bilateral operator weighted shifts

“…In [4], Ben-Artzi and Gohberg introduced the concepts of Bohl exponent and canonical splitting projection to describe the spectrum and the essential spectrum of operator weighted shifts of finite multiplicity (see also [10]). However, in the general setting of bilateral operator weighted shifts of infinite multiplicity, the complete description of the spectrum and its parts is not yet settled.…”