Abstract
A truncated Toeplitz operator is a compression of the multiplicationoperator to a backward shift invariant subspace of the Hardy space H^2. Anasymmetric truncated Toeplitz operator is a compression of the multiplication operator that acts between two different backward shift invariant subspaces of H^2. All rank-one truncated Toeplitz operators have been described by Sarason. Here, we characterize all rank-one asymmetric truncated Toeplitz operators. This completes the description given by Łanucha for asymmetric truncated Toeplitz operators on finite-dimensional backward shift invariant subspaces.
Highlights
1 Introduction Denote by H 2 the Hardy space of the open unit disk D = {z : |z| < 1} and let P be the orthogonal projection from L2(∂D) onto H 2
A classical Toeplitz operator Tφ with symbol φ ∈ L∞(∂D) is defined on H 2 by
It is clear that a Toeplitz operator with symbol from L∞(∂D) is bounded
Summary
The authors in [8] showed that the operators kwβ ⊗ kwα and kwβ ⊗ kwα belong to T (α, β) for every w ∈ D and every w ∈ ∂D such that both α and β have an ADC at w. We prove that if both Kα and Kβ have dimension larger than one (and not necessarily finite), the only rank-one operators in T (α, β) are the non-zero scalar multiples of kwβ ⊗ kwα and kwβ ⊗ kwα .
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