Asymptotic Symbols and Fredholm Theory for Variable-Coefficient Finite-Band Norlund-Type Operators on \(c_0\) and c
Moses Kwabena Yeboah *
Department of Mathematics, University for Development Studies, Tamale, Ghana.
*Author to whom correspondence should be addressed.
Abstract
This paper develops an operator-theoretic framework for a class of variable-coefficient finite-band N¨orlund-type averaging operators acting on the classical Banach sequence spaces c0 and c. Instead of assuming that the matrix becomes exactly translation invariant after finitely many rows, the coefficients are allowed to vary indefinitely, subject only to convergence toward a limiting finite band. For a fixed integer m ≥ 1, we consider \[(\mathcal{N} x)_n=\sum_{j=0}^{\min \{m, n\}} a_{j, n} x_{n-j}, \quad a_{j, n} \longrightarrow a_j,\] and associate with N the asymptotic polynomial symbol
\(a(z)=a_0+a_1 z+\cdots+a_m z^m .\)
The first main result is a compactness characterization: the convergence of the band coefficients is equivalent to compact equivalence between N and the shift polynomial a(s). The resulting perturbation is generally compact of infinite rank, so the framework strictly contains the eventually constant finite-band case. Using Atkinson’s theorem, the Calkin algebra, polynomial spectral mapping, and a direct Fredholm analysis of the unilateral right shift, it is proven that
\(\sigma_{\mathrm{ess}}(\mathcal{N})=a(\mathbb{T})\)
on both c0 and c. If \(\lambda\) ∉ a(T) and Na (\(\lambda\)) denotes the number, counted with multiplicity, of zeros of a(z) − \(\lambda\) in the open unit disk, then
\(\operatorname{ind}(\mathcal{N}-\lambda I)=-N_a(\lambda)=-\operatorname{wind}(a(\mathbb{T}), \lambda) .\)
We further obtain a localization theorem for the full spectrum,
\(a(\overline{\mathbb{D}}) \subseteq \sigma(\mathcal{N}) \subseteq a(\overline{\mathbb{D}}) \cup\left\{a_{0, n}: n \geq 0\right\},\)
and consequently an exact formula \(\sigma\)(N) = a(\(\bar{D}\)) whenever every diagonal coefficient a0,n lies in a(\(\bar{D}\)). For normalized nonnegative averaging coefficients, the limiting symbol satisfies a(1) = 1, the essential spectral radius is one, and the constant sequence is an embedded eigenvector at \(\lambda\) = 1 on c. Two variable-coefficient examples are developed in detail, including a three-term family whose difference from its limiting Toeplitz operator is compact but of infinite rank.
Keywords: Norlund-type operator, Fredholm index, compact perturbation, asymptotic symbol, essential spectrum, unilateral shift