The Essential Spectrum and Banach Reducibility of Operator Weighted Shifts

Cowen-Douglas Operator and Shift on Basis

“…A typical example for a n-multiplicity Cowen-Douglas operator is n-multiplicity backward shift on an orthonormal basis. A characterization n-multiplicity backward operator weighted shifts being Cowen-Douglas operators has ever given in terminology of their weight sequences [9]. In this paper, we shall show the following.…”

Fine Structure of the Dichotomy Spectrum

“…In this situation σ(T A ) becomes a solid disc, whereas the essential spectrum σ F (T A ) inherits the above structure as union of rings and is therefore invariant under compact perturbations. This is shown in [9] or [32] by means of Fredholm properties of the shift operator…”

“…Unitary equivalence for T + A is studied in [28], local spectral theory for matrixweighted shifts was investigated in [30] and finally, [32] characterize Banach reducibility. Exponential dichotomies.…”

“…• In the operator theoretical literature [7][8][9]11,12,28,30,32] the weighted shift operators T A and T + A are typically studied on the Hilbert space of square-summable sequences 2 . To a large extend, we instead focus on the Banach space ∞ of bounded sequences.…”

On unitary equivalence of bilateral operator valued weighted shifts

“…The are some partial results regarding unitary equivalence of bilateral operator valued weighted shifts. Li, Ji and Sun proved that each bilateral operator valued weighted shift with invertible weights defined on C m for m ≥ 2 is unitarily equivalent to a shift with upper triangular weights (see [12,Theorem 2.1]). Shields provided in [15] characterization of unitary equivalence in case of classical bilateral shifts.…”