Asymmetric truncated Toeplitz operators equal to the zero operator

Abstract

Asymmetric truncated Toeplitz operators are compressions of multiplication operators acting between two model spaces. These operators are natural generalizations of truncated Toeplitz operators. In this paper we describe symbols of asymmetric truncated Toeplitz operators equal to the zero operator.

Asymmetric truncated Toeplitz operators on finite-dimensional spaces

“…Proof of Proposition 2.1. By [14,Cor. 2.6], every operator A in T (α, β) can be written as a sum A = A χ + A ψ with χ ∈ K α and ψ ∈ K β .…”

“…In the proof of Proposition 2.1 we use the fact that if α, β are two nonconstant inner functions, then A α,β ϕ = 0 if and only if ϕ ∈ αH 2 + βH 2 (see [14, Thm 2.1] for proof). We also use the following simple lemma from [14]. Proof of Proposition 2.1.…”

Compressions of kth-order slant Toeplitz operators to model spaces

“…Observe that U α,β ϕ can be seen as a composition of W k and an asymmetric truncated Toeplitz operator. Indeed, using Lemma 1(g), for f ∈ K ∞ α , we get [4,22]). Let α be an inner function.…”

Asymmetric Truncated Hankel Operators: Rank One, Matrix Representation

“…This paper determines all rank one asymmetric truncated Hankel operators and generalizes the results about matrix representation to ATHOs and ATTOs on special model spaces. In Section 2, we cite some results from [3,[10][11][12]. We precise all rank one ATHOs in Section 3.…”