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.
Highlights
As usual, let H 2 denote the classical Hardy space
An asymmetric truncated Toeplitz operator Aαφ,β with symbol φ ∈ L2 is an operator from Kα into Kβ densely defined by
An asymmetric truncated Hankel operator Bφα,β with symbol φ ∈ L2 is an operator from Kα into Kβ densely defined by
Summary
Let H 2 denote the classical Hardy space. The space H 2 can be seen as a space of functions analytic in the unit disk D = {z : |z| < 1} or as a closed subspace of L2 = L2(∂D). A truncated Toeplitz operator Aαφ with symbol φ ∈ L2 is densely defined on the model space Kα by. An asymmetric truncated Toeplitz operator Aαφ,β with symbol φ ∈ L2 is an operator from Kα into Kβ densely defined by. An asymmetric truncated Hankel operator Bφα,β with symbol φ ∈ L2 is an operator from Kα into Kβ densely defined by. Sarason [12] proved that a bounded linear operator A on Kα is a truncated Toeplitz operator if and only if A − Sα∗ ASα is an operator of a special kind and rank at most two. Gu showed in [7] that a bounded linear operator B on Kα is a truncated Hankel operator if and only if B Sα − Sα∗ B is a special kind of operator of rank at most two.
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