Kernels of Unbounded Toeplitz Operators and Factorization of Symbols

Câmara, M. Cristina; Malheiro, Teresa; Partington, Jonathan R.

doi:10.1007/s00025-020-01323-z

Cited by 5 publications

(7 citation statements)

References 19 publications

(35 reference statements)

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“…Remark 6.3. The relations between ker T G and ker T βG were studied in [5] where it was shown, in particular, that ker…”

Section: Toeplitz Operatorsmentioning

confidence: 99%

“…dim ker T G . However, it may be difficult to see whether or not ker T βG = {0} when ker T G is infinite dimensional and β is not a finite Blaschke product (see for instance [5] for some examples).…”

Section: Toeplitz Operatorsmentioning

confidence: 99%

“…It was shown in [9] that every Toeplitz kernel has a maximal function and, in [5,8] that f M is a maximal function for ker T G if and only if Gf M = zh where h ∈ H 2 is outer. We have…”

Section: Toeplitz Operatorsmentioning

confidence: 99%

“…It was shown in [7] that every Toeplitz kernel has a maximal function and, in [5,8] that f M is a maximal function for ker T G if and only if G f M = z h, where h ∈ H 2 is outer. We have Therefore, P ker TG (z Ḡ) is a maximal function for ker T G if and only if P ker TG 1 is outer.…”

Section: Proposition 62 Let β Be An Inner Function and Let Gmentioning

confidence: 99%

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General Toeplitz kernels and -invariance

CÂMARA¹,

Kliś–Garlicka²,

Ptak³

2023

Can. J. Math.-J. Can. Math.

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Motivated by the near invariance of model spaces for the backward shift, we introduce a general notion of (X, Y )-invariant operators. The relations between this class of operators and the near invariance properties of their kernels are studied. Those lead to orthogonal decompositions for the kernels, which generalize well known orthogonal decompositions of model spaces. Necessary and sufficient conditions for those kernels to be nearly X-invariant are established. This general approach can be applied to a wide class of operators defined as compressions of multiplication operators, in particular to Toeplitz operators and truncated Toeplitz operators, to study the invariance properties of their kernels (general Toeplitz kernels).

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“…Remark 6.3. The relations between ker T G and ker T βG were studied in [5] where it was shown, in particular, that ker…”

Section: Toeplitz Operatorsmentioning

confidence: 99%

Section: Toeplitz Operatorsmentioning

confidence: 99%

“…It was shown in [9] that every Toeplitz kernel has a maximal function and, in [5,8] that f M is a maximal function for ker T G if and only if Gf M = zh where h ∈ H 2 is outer. We have…”

Section: Toeplitz Operatorsmentioning

confidence: 99%

Section: Proposition 62 Let β Be An Inner Function and Let Gmentioning

confidence: 99%

See 2 more Smart Citations

General Toeplitz kernels and -invariance

CÂMARA¹,

Kliś–Garlicka²,

Ptak³

2023

Can. J. Math.-J. Can. Math.

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“…2] to T ω . Note also that in [11] one finds a detailed study of the kernel of M + (κ). Thus one has the following properties.…”

Section: (A) Some General Properties Of Unbounded Tomentioning

confidence: 99%

Unbounded Toeplitz Operators with Rational Symbols

Castrigiano¹

2021

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Unbounded (and bounded) Toeplitz operators (TO) with rational symbols are analysed in detail showing that they are densely defined closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains, ranges, spectral and Fredholm points are determined. In particular, in the symmetric case, i.e., for a real rational symbol the deficiency spaces and indices are explicitly available. -The concluding section gives a brief overview on the research on unbounded TO in order to locate the present contribution. Regarding properties of unbounded TO in general, it furnishes some new results recalling the close relationship to Wiener-Hopf operators and, in case of semiboundedness, to singular operators of Hilbert transformation type. Specific symbols considered in the literature admit further analysis. Some conclusions are drawn for semibounded integrable and real square-integrable symbols. There is an approach to semibounded TO, which starts from closable semibounded forms related to a Toeplitz matrix. The Friedrichs extension of the TO associated with such a form is studied. Finally, analytic TO and Toeplitz-like operators are briefly examined, which in general differ from the TO treated here.

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Paired Kernels and Their Applications

Câmara,

Partington

2024

Results Math

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This paper considers paired operators in the context of the Lebesgue Hilbert space on the unit circle and its subspace, the Hardy space $$H^2$$ H 2 . The kernels of such operators, together with their analytic projections, which are generalizations of Toeplitz kernels, are studied. Results on near-invariance properties, representations, and inclusion relations for these kernels are obtained. The existence of a minimal Toeplitz kernel containing any projected paired kernel and, more generally, any nearly $$S^*$$ S ∗ -invariant subspace of $$H^2$$ H 2 , is derived. The results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators.

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Kernels of Unbounded Toeplitz Operators and Factorization of Symbols

Abstract: We consider kernels of unbounded Toeplitz operators in $$H^p({\mathbb {C}}^{+})$$ H p ( C + ) in terms of a factorization of their symbols. We… Show more

Cited by 5 publications

References 19 publications

General Toeplitz kernels and -invariance

General Toeplitz kernels and -invariance

Unbounded Toeplitz Operators with Rational Symbols

Paired Kernels and Their Applications

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