Algebraic Properties of Toeplitz operators

Brown, Arlen; Halmos, Paul R.

doi:10.1007/978-1-4613-8208-9_19

Cited by 222 publications

(334 citation statements)

References 5 publications

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“…Brown and P.R. Halmos [3], and so it is somewhat of a surprise that 25 years passed before the exact nature of the relationship between the symbol ϕ ∈ L ∞ and the positivity of the selfcommutator [T * ϕ , T ϕ ] was understood (via Cowen's theorem [7]). As Cowen notes in his survey paper [6], the intensive study of subnormal Toeplitz operators in the 1970's and early 80's is one explanation for the relatively late appearence of the sequel to the Brown-Halmos work.…”

Section: Hyponormality Of Toeplitz Operators With Trigonometric Polynmentioning

confidence: 99%

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Hyponormality and spectra of Toeplitz operators

Farenick

Lee

1996

Trans. Amer. Math. Soc.

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Abstract. This paper concerns algebraic and spectral properties of Toeplitz operators Tϕ, on the Hardy space H 2 (T), under certain assumptions concerning the symbols ϕ ∈ L ∞ (T). Among our algebraic results is a characterisation of normal Toeplitz opertors with polynomial symbols, and a characterisation of hyponormal Toeplitz operators with polynomial symbols of a prescribed form. The results on the spectrum are as follows. It is shown that by restricting the spectrum, a set-valued function, to the set of all Toeplitz operators, the spectrum is continuous at Tϕ, for each quasicontinuous ϕ. Secondly, we examine under what conditions a classic theorem of H. Weyl, which has extensions to hyponormal and Toeplitz operators, holds for all analytic functions of a single Toeplitz operator with continuous symbol.

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Section: Hyponormality Of Toeplitz Operators With Trigonometric Polynmentioning

confidence: 99%

“…and hence The following necessary and sufficient condition for normality is to be expected, given that every normal Toeplitz operator is a translation and rotation of a hermitian Toeplitz operator [3]. …”

Section: The Following Equation In C M Holdsmentioning

confidence: 99%

Hyponormality and spectra of Toeplitz operators

Farenick

Lee

1996

Trans. Amer. Math. Soc.

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“…Theorem 3.1 and Corollary 3.2 are natural analogs of results established for Toeplitz operators in the classical setting (see [4]). …”

Section: Toeplitz Operatorsmentioning

confidence: 64%

Wiener Algebra for the Quaternions

et al. 2015

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“…As is well known, for f, g ∈ L ∞ , Brown and Halmos [2] have shown that the product of two Toeplitz operators T f and T g is also a Toeplitz operator if and only iff ∈ H ∞ or g ∈ H ∞ . Axler et al [1] have shown that the product T f T g is a finiterank perturbation of a Toeplitz operator if and only if one of the operators Hf or H g has finite rank, where f, g both are in L ∞ .…”

Section: Consequencesmentioning

confidence: 96%

The Finite Sum of the Products of Two Toeplitz Operators

Ding¹

2009

J. Aust. Math. Soc.

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We consider in this paper the question of when the finite sum of products of two Toeplitz operators is a finite-rank perturbation of a single Toeplitz operator on the Hardy space over the unit disk. A necessary condition is found. As a consequence we obtain a necessary and sufficient condition for the product of three Toeplitz operators to be a finite-rank perturbation of a single Toeplitz operator.2000 Mathematics subject classification: primary 47B35.

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Algebraic Properties of Toeplitz operators

Cited by 222 publications

References 5 publications

Hyponormality and spectra of Toeplitz operators

Hyponormality and spectra of Toeplitz operators

Wiener Algebra for the Quaternions

The Finite Sum of the Products of Two Toeplitz Operators

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