Compressions of kth-order slant Toeplitz operators to model spaces

Abstract

In this paper, we consider compressions of kth-order slant Toeplitz operators to the backward shift-invariant subspaces of the classical Hardy space H2. In particular, we characterize these operators using compressed shifts and finite-rank operators of special kind.

Products of compressions of $k^{th}$-order slant Toeplitz operators to model spaces

“…and (a) follows. Part (b) follows from Lemma 2(a) in [31] and the fact that for f ∈ K α we have (S * α ) m f = (S * ) m f . Recall that for f ∈ K α we have Sf = zf ∈ K α if and only if f ⊥ k α 0 .…”

“…As in the case of truncated Toeplitz operators, the proof is based on a characterization of operators from S k (α, β) in terms of operators S α and S β . It was proved in [31] that a bounded linear operator U : K α → K β belongs to S k (α, β) if and only if there exist χ ∈ K α and ψ 0 , . .…”

“…Let α be a nonconstant inner function, let m ∈ N and let f ∈ K α . Then (a) z m f (z) ∈ K α if and only if [31,Cor. 6] there are χ ∈ K α and ψ 0 , .…”

“…(a) Assume that A β,γ ϕ ∈ T (β, γ) and [31]). It follows that every operator from S k (α, β) has a symbol of the form ϕ − + ϕ + , where…”

“…The class S k (α, β) was first introduced in [31] where some basic algebraic properties of its elements were investigated. Here we focus on some commuting relations for operators from S k (α, β) and on products of this kind of operators.…”