Hyponormality of Toeplitz Operators with Polynomial Symbols: An Extremal Case

Hwang, In SungHwang; Kim, In Hyoun; Lee, Woo Young

doi:10.1002/1522-2616(200111)231:1<25::aid-mana25>3.0.co;2-x

Cited by 10 publications

(4 citation statements)

References 11 publications

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“…Halmos [BH]. The exact nature of the relationship between the symbol ϕ ∈ L ∞ and the hyponormality of the Toeplitz operator T ϕ was understood in 1988 via Cowen's Theorem [Co2] -this elegant and useful theorem has been used in the works [CuL1], [CuL2], [FL], [Gu1], [Gu2], [GS], [HKL1], [HKL2], [HL1], [HL2], [HL3], [Le], [NT], [Zhu], and others; these works have been devoted to the study of hyponormality for Toeplitz operators on H 2 . Particular attention has been paid to Toeplitz operators with polynomial symbols or rational symbols [HL1], [HL2], [HL3].…”

Section: Introductionmentioning

confidence: 99%

Matrix Functions of Bounded Type: An Interplay Between Function Theory and Operator Theory

2019

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In this paper, we study matrix functions of bounded type from the viewpoint of describing an interplay between function theory and operator theory. We first establish a criterion on the coprime-ness of two singular inner functions and obtain several properties of the Douglas-Shapiro-Shields factorizations of matrix functions of bounded type. We propose a new notion of tensored-scalar singularity, and then answer questions on Hankel operators with matrix-valued bounded type symbols. We also examine an interpolation problem related to a certain functional equation on matrix functions of bounded type; this can be seen as an extension of the classical Hermite-Fejér Interpolation Problem for matrix rational functions. We then extend the H ∞ -functional calculus to an H ∞ + H ∞ -functional calculus for the compressions of the shift. Next, we consider the subnormality of Toeplitz operators with matrix-valued bounded type symbols and, in particular, the matrixvalued version of Halmos's Problem 5; we then establish a matrix-valued version of Abrahamse's Theorem. We also solve a subnormal Toeplitz completion problem of 2 × 2 partial block Toeplitz matrices. Further, we establish a characterization of hyponormal Toeplitz pairs with matrix-valued bounded type symbols, and then derive rank formulae for the self-commutators of hyponormal Toeplitz pairs.

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Section: Introductionmentioning

confidence: 99%

Matrix Functions of Bounded Type: An Interplay Between Function Theory and Operator Theory

2019

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“…This elegant and useful theorem has been used in [CuL1], [CuL2], [FL], [Gu1], [Gu2], [GS], [HKL1], [HKL2], [HL1], [HL2], [HL3], [Le], [NT] and [Zhu], which have been devoted to the study of hyponormality for Toeplitz operators on H 2 . When one studies the hyponormality (also, normality and subnormality) of the Toeplitz operator T ϕ one may, without loss of generality, assume that ϕ(0) = 0; this is because hyponormality is invariant under translation by scalars.…”

Section: Introductionmentioning

confidence: 99%

Abrahamse's Theorem for matrix-valued symbols and subnormal Toeplitz completions

Curto,

Hwang,

Lee

2013

Preprint

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This paper deals with subnormality of Toeplitz operators with matrix-valued symbols and, in particular, with an appropriate reformulation of Halmos's Problem 5: Which subnormal Toeplitz operators with matrix-valued symbols are either normal or analytic ? In 1976, M. Abrahamse showed that if ϕ ∈ L ∞ is such that ϕ or ϕ is of bounded type and if Tϕ is subnormal, then Tϕ is either normal or analytic. In this paper we establish a matrix-valued version of Abrahamse's Theorem and then apply this result to solve the following Toeplitz completion problem: Find the unspecified Toeplitz entries of the partial block Toeplitz matrixso that A becomes subnormal, where b λ is a Blaschke factor of the form b λ (z) := z−λ 1−λz (λ ∈ D).

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“…This elegant and useful theorem has been used in the works [CuL1], [CuL2], [FL], [Gu1], [Gu2], [GS], [HKL1], [HKL2], [HL1], [HL2], [HL3], [Le], [NT] and [Zhu], which have been devoted to the study of hyponormality for Toeplitz operators on H 2 . Particular attention has been paid to Toeplitz operators with polynomial symbols or rational symbols [HL2], [HL3].…”

Section: Introductionmentioning

confidence: 99%

Hyponormality and subnormality of block Toeplitz operators

Curto¹,

Hwang²,

Lee³

2012

Advances in Mathematics

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In this paper we are concerned with hyponormality and subnormality of block Toeplitz operators acting on the vector-valued Hardy space H 2 C n of the unit circle. Firstly, we establish a tractable and explicit criterion on the hyponormality of block Toeplitz operators having bounded type symbols via the triangularization theorem for compressions of the shift operator.Secondly, we consider the gap between hyponormality and subnormality for block Toeplitz operators. This is closely related to Halmos's Problem 5: Is every subnormal Toeplitz operator either normal or analytic ? We show that if Φ is a matrix-valued rational function whose coanalytic part has a coprime factorization then every hyponormal Toeplitz operator TΦ whose square is also hyponormal must be either normal or analytic.Thirdly, using the subnormal theory of block Toeplitz operators, we give an answer to the following "Toeplitz completion" problem: Find the unspecified Toeplitz entries of the partial block Toeplitz matrix A := U * ? ?U * so that A becomes subnormal, where U is the unilateral shift on H 2 .

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Hyponormality of Toeplitz Operators with Polynomial Symbols: An Extremal Case

Cited by 10 publications

References 11 publications

Matrix Functions of Bounded Type: An Interplay Between Function Theory and Operator Theory

Matrix Functions of Bounded Type: An Interplay Between Function Theory and Operator Theory

Abrahamse's Theorem for matrix-valued symbols and subnormal Toeplitz completions

Hyponormality and subnormality of block Toeplitz operators

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