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.
Highlights
Let T be the unit circle in the complex plane C
We start by recalling that the Hilbert space L2 = L2(T) is the space of all square-integrable functions on the unit circle T equipped with the normalized Lebesgue measure dm(eiθ )
An orthonormal basis of L2 is given by the set {en(θ) : n ∈ Z}, where en(θ) = einθ for θ ∈ R
Summary
Let L∞ be the Banach space of essentially bounded functions on T. For any φ ∈ L∞ , the bounded multiplication operator Mφ is defined by the formula. The Hardy space of the circle H2 is the set of functions f ∈ L2 such that fn = 0 for all n < 0 , and let. We introduce an important class of operators on spaces of analytic functions, which is the class of Toeplitz operators. Given that φ ∈ L∞ , the Toeplitz operator Tφ : H2 → H2 is defined by. Note that the Toeplitz operator becomes bounded if and only if φ ∈ L∞. In this case, we have ∥Tφ∥ = ∥φ∥∞ (see [1]).
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