Invertibility, Fredholmness and kernels of dual truncated Toeplitz operators

Câmara, M. Cristina; Kliś–Garlicka, Kamila; Łanucha, Bartosz; Ptak, Marek

doi:10.1007/s43037-020-00077-8

Cited by 13 publications

(14 citation statements)

References 16 publications

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“…Theorem 6.6 in [8] shows that for a symbol g that is invertible in L ∞ we have ker D θ g = g −1 ker A θ g −1 , so given our observation (7) under the condition that g is invertible in L ∞ we can write ker D θ g as an L 2 function multiplied by a model space. We now aim to use similar recursive methods that were used to prove Theorem 3.4 to obtain a decomposition theorem for ker D θ g .…”

Section: Application To Dual Truncated Toeplitz Operatorsmentioning

confidence: 85%

“…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]. Although the domain of the dual truncated Toeplitz operator is not a subspace of H 2 we still can use similar recursive techniques used in previous sections to decompose the the kernel in to a fixed function multiplied by a S * -invariant subspace.…”

Section: Definition 12 the Truncated Toeplitz Operator A θmentioning

confidence: 99%

“…If ker D θ g ⊆ C then ker D θ g = {0}.norm , and so in the L 1 norm we must have g f = lim Now two applications of Hölder's inequality shows theL 1 limit of N i=0 g f 0 λ i z i + N j=0 gh 0 μ j z j is equal to g f 0 ∞ i=0 λ i z i + gh 0 ∞ j=0 μ j z j , where ∞ i=0 λ i z i , ∞j=0 μ j z j are limits in the H 2 sense . So we may writeg f = g f 0 ∞ i=0 λ i z i + gh 0 ∞ j=0 μ j z j ,and furthermore by taking limits in(8) we can deduce||g f || 2 H 2 = ∞ i=0 |λ i | 2 + ∞ i=0Mimicking the argument from Sect. 3 between (5) and(6)we can say f ∈ ker D θ g if and only if g f = g f 0 gh 0 closed S * -invariant subspace of H 2 (D, C 2 ).…”

mentioning

confidence: 95%

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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: Application To Dual Truncated Toeplitz Operatorsmentioning

confidence: 85%

Section: Definition 12 the Truncated Toeplitz Operator A θmentioning

confidence: 99%

mentioning

confidence: 95%

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Nearly Invariant Subspaces with Applications to Truncated Toeplitz Operators

O'Loughlin

2020

Complex Anal. Oper. Theory

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“…where P − = I − P + . Section 5 of [5] shows the dual truncated Toeplitz operator is EAE to a paired operator on (L 2 ) 2 . Throughout this section, unless otherwise stated, we assume that G ∈ L (p,n×n) where p ∈ (2, ∞].…”

Section: The Relationmentioning

confidence: 99%

Matrix-Valued Truncated Toeplitz Operators: Unbounded Symbols, Kernels and Equivalence After Extension

O'Loughlin

2022

Integr. Equ. Oper. Theory

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This paper studies matrix-valued truncated Toeplitz operators, which are a vectorial generalisation of truncated Toeplitz operators. It is demonstrated that, although there exist matrix-valued truncated Toeplitz operators without a matrix symbol in $$L^p$$ L p for any $$p \in (2, \infty ]$$ p ∈ ( 2 , ∞ ] , there is a wide class of matrix-valued truncated Toeplitz operators which possess a matrix symbol in $$L^p$$ L p for some $$p \in (2, \infty ]$$ p ∈ ( 2 , ∞ ] . In the case when the matrix-valued truncated Toeplitz operator has a symbol in $$L^p$$ L p for some $$p \in (2, \infty ]$$ p ∈ ( 2 , ∞ ] , an approach is developed which bypasses some of the technical difficulties which arise when dealing with problems concerning matrix-valued truncated Toeplitz operators with unbounded symbols. Using this new approach, two new notable results are obtained. The kernel of the matrix-valued truncated Toeplitz operator is expressed as an isometric image of an $$S^*$$ S ∗ -invariant subspace. Also, a Toeplitz operator is constructed which is equivalent after extension to the matrix-valued truncated Toeplitz operator. In a different yet overlapping vein, it is also shown that multidimensional analogues of the truncated Wiener–Hopf operators are unitarily equivalent to certain matrix-valued truncated Toeplitz operators.

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“…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]).…”

Section: Introductionmentioning

confidence: 99%

Compressions of Multiplication Operators and Their Characterizations

et al. 2020

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Dual truncated Toeplitz operators and other restrictions of the multiplication by the independent variable $$M_z$$ M z on the classical $$L^2$$ L 2 space on the unit circle are investigated. Commutators are calculated and commutativity is characterized. A necessary and sufficient condition for any operator to be a dual truncated Toeplitz operator is established. A formula for recovering its symbol is stated.

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Invertibility, Fredholmness and kernels of dual truncated Toeplitz operators

Cited by 13 publications

References 16 publications

Nearly Invariant Subspaces with Applications to Truncated Toeplitz Operators

Nearly Invariant Subspaces with Applications to Truncated Toeplitz Operators

Matrix-Valued Truncated Toeplitz Operators: Unbounded Symbols, Kernels and Equivalence After Extension

Compressions of Multiplication Operators and Their Characterizations

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