Dual truncated Toeplitz operators

Ding, Xin; Sang, Yuanqi

doi:10.1016/j.jmaa.2017.12.032

Cited by 26 publications

(21 citation statements)

References 13 publications

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“…5 we study the kernel of dual truncated Toeplitz operator. Dual truncated Toeplitz operators have been studied in both [9,11] as well as many other sources. The kernel of a dual truncated Toeplitz operator has been studied in [8].…”

Section: Definition 12 the Truncated Toeplitz Operator A θmentioning

confidence: 99%

Nearly Invariant Subspaces with Applications to Truncated Toeplitz Operators

O'Loughlin

2020

Complex Anal. Oper. Theory

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In this paper we first study the structure of the scalar and vector-valued nearly invariant subspaces with a finite defect. We then subsequently produce some fruitful applications of our new results. We produce a decomposition theorem for the vector-valued nearly invariant subspaces with a finite defect. More specifically, we show every vector-valued nearly invariant subspace with a finite defect can be written as the isometric image of a backwards shift invariant subspace. We also show that there is a link between the vector-valued nearly invariant subspaces and the scalar-valued nearly invariant subspaces with a finite defect. This is a powerful result which allows us to gain insight in to the structure of scalar subspaces of the Hardy space using vector-valued Hardy space techniques. These results have far reaching applications, in particular they allow us to develop an all encompassing approach to the study of the kernels of: the Toeplitz operator, the truncated Toeplitz operator, the truncated Toeplitz operator on the multiband space and the dual truncated Toeplitz operator.

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Section: Definition 12 the Truncated Toeplitz Operator A θmentioning

confidence: 99%

Nearly Invariant Subspaces with Applications to Truncated Toeplitz Operators

O'Loughlin

2020

Complex Anal. Oper. Theory

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“…We start with some elementary properties of asymmetric dual truncated Toeplitz operators. These properties were proved in [9]…”

Section: Elementary Propertiesmentioning

confidence: 90%

“…It is natural to consider dual truncated Toeplitz operators, defined analogously as compressions of multiplication operators to the orthogonal complement of a model space in L 2 ( ) . These operators were very recently introduced and studied in [9,13,15]. It turns out that, in this case, they behave very differently from truncated Toeplitz operators.…”

Section: Introductionmentioning

confidence: 99%

Invertibility, Fredholmness and kernels of dual truncated Toeplitz operators

Câmara

Kliś–Garlicka

Łanucha

et al. 2020

Banach J. Math. Anal.

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Asymmetric dual truncated Toeplitz operators acting between the orthogonal complements of two (eventually different) model spaces are introduced and studied. They are shown to be equivalent after extension to paired operators on L 2 () ⊕ L 2 () and, if their symbols are invertible in L ∞ () , to asymmetric truncated Toeplitz operators with the inverse symbol. Relations with Carleson's corona theorem are also established. These results are used to study the Fredholmness, the invertibility and the spectra of various classes of dual truncated Toeplitz operators.

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“…An orthonormal basis of L 2 is given by the set {e n (θ) : n ∈ Z} , where e n (θ) = e inθ for θ ∈ R . The For all f, g ∈ L 2 , the tensor product f ⊗ g is the rank one operator in L 2 and is defined by…”

Section: Introductionmentioning

confidence: 99%