Dual-Band General Toeplitz Operators

Abstract

We relate dual-band general Toeplitz operators to block truncated Toeplitz operators and, via equivalence after extension, with Toeplitz operators with $$4 \times 4$$ 4 × 4 matrix symbols. We discuss their norm, their kernel, Fredholmness, invertibility and spectral properties in various situations, focusing on the spectral properties of the dual-band shift, which turns out to be considerably complex, leading to new and nontrivial connections with the boundary behaviour of the associated inner function.

General Toeplitz kernels and -invariance

“…If K = H we write T H ϕ . These are particular cases of the so-called general Wiener-Hopf operators [1,13,31], which we call general Toeplitz operators [6].…”

“…and β ḠK 6 If G = ᾱ, where α is an inner function, then ker T G = K 2 α . If β < α, the decompositions (6.4) and (6.5) become…”

“…These results allow us to establish necessary and sufficient conditions for those kernels to be nearly X -invariant, with or without defect. They also allow for a general approach to the study of a wide class of operators defined as compressions of multiplications operators (general Toeplitz operators [6]) and the invariance properties of their kernels (general Toeplitz kernels). In Sections 5 and 6, we apply those results to Toeplitz operators and truncated Toeplitz operators.…”