Characterizations of Asymmetric Truncated Toeplitz and Hankel Operators

Abstract

It was recently proved that in some special cases asymmetric truncated Toeplitz operators can be characterized in terms of compressed shifts and rank-two operators of special form. In this paper we show that such characterizations hold in all cases. We also show a connection between asymmetric truncated Toeplitz operators and asymmetric truncated Hankel operators. We use this connection to generalize results known for truncated Hankel operators to asymmetric truncated Hankel operators.

Matrix Representations of Asymmetric Truncated Toeplitz Operators

“…for all 1 ≤ p ≤ m and 1 ≤ s ≤ n. Since A ∈ T (α, β) if and only if A * ∈ T (β, α) [14], the next theorem follows easily from (3.6) and (3.7).…”

“…We refer the reader to the survey [10] for more results and references (see also [9]). A natural generalization of truncated Toeplitz operators, asymmetric truncated Toeplitz operators, were introduced more recently in [3,4] and [14].…”

“…Recall that (A α,β ϕ ) * = A β,α ϕ [14]. To find a symbol for A ∈ T (α, β) it is therefore enough to find a symbol for A * ∈ T (β, α).…”

Asymmetric Truncated Hankel Operators: Rank One, Matrix Representation

“…One of the most important results about TTOs and THOs is their characterization in terms of compressed shift operator and operators of rank at most 2. Recently, the authors in [10] proved similar characterizations for both ATTO and ATHO.…”

“…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.…”