On the product of dilation of truncated Toeplitz operators

Abstract

In this paper we study when the product of two dilations of truncated Toeplitz operators gives a dilation of a truncated Toeplitz operator. We will use an approach established in a recent paper written by Ko and Lee. This approach allows us to represent the dilation of the truncated Toeplitz operator via a 2 × 2 block operator.

Compressions of Multiplication Operators and Their Characterizations

“…The proof of part (2) given in [17] was based on the fact that B θ ϕ can be expressed as a block operator built from certain products of classical Hankel operators, and only for ϕ ∈ L ∞ . But (2) can be proved in an alternative, simpler way.…”

“…Systematic study of truncated Toeplitz operators A θ ϕ (for general ϕ ∈ L 2 ) was started in [22] while the properties of dual truncated Toeplitz operators D θ ϕ were investigated in [8,15,20] and more recently in [2,6,7,21]. Truncated Hankel operators were studied in [17] and also in [14], but there is a different definition of B θ ϕ (see also [3]).…”