Compact composition operators on Bergman-Orlicz spaces

Lefèvre, Pascal; Li, Daniel; Queffélec, Hervé; Rodrı́guez-Piazza, Luis

doi:10.1090/s0002-9947-2013-05922-3

Cited by 29 publications

(46 citation statements)

References 19 publications

(23 reference statements)

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“…where the infimum is over the Blaschke products with less than n zeros (in the Hilbert space H 2 , the approximation number a n (C ϕ ) is equal to the Gelfand number c n (C ϕ ), which is, by definition, less or equal to C ϕ |BH 2 , since BH 2 is of codimension < n). This map was first introduced in [11] (see also [13]). Explicitly, ϕ is defined as follows.…”

Section: (Using the Intermediate Value Theorem)mentioning

confidence: 99%

Estimates for approximation numbers of some classes of composition operators on the Hardy space

Li¹,

Queffélec²,

Rodrı́guez-Piazza³

2013

Ann. Acad. Sci. Fenn. Math.

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Abstract. We give estimates for the approximation numbers of composition operators on H 2 , in terms of some modulus of continuity. For symbols whose image is contained in a polygon, we get that these approximation numbers are dominated by e −c √ n . When the symbol is continuous on the closed unit disk and has a domain touching the boundary non-tangentially at a finite number of points, with a good behavior at the boundary around those points, we can improve this upper estimate. A lower estimate is given when this symbol has a good radial behavior at some point.As an application we get that, for the cusp map, the approximation numbers are equivalent, up to constants, to e −cn/ log n , very near to the minimal value e −cn . We also see the limitations of our methods. To finish, we improve a result of El-Fallah, Kellay, Shabankhah and Youssfi, in showing that for every compact set K of the unit circle T with Lebesgue measure 0, there exists a compact composition operator C φ : H 2 → H 2 , which is in all Schatten classes, and such that φ = 1 on K and |φ| < 1 outside K.

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Section: (Using the Intermediate Value Theorem)mentioning

confidence: 99%

Estimates for approximation numbers of some classes of composition operators on the Hardy space

Li¹,

Queffélec²,

Rodrı́guez-Piazza³

2013

Ann. Acad. Sci. Fenn. Math.

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“…We first recall the context of this work, which appears as a continuation of [9], [10], [11], [14] and [15].…”

Section: Introductionmentioning

confidence: 99%

Some new properties of composition operators associated with lens maps

Lefèvre

Queffélec

et al. 2012

Isr. J. Math.

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“…The next theorem, stated for ψ in the ∆ 2 -class, is a particular case of the one obtained in [11,14] for a larger class of Orlicz functions (see also [25,28] for the unit disc).…”

Section: Composition Operators On Weightedmentioning

confidence: 91%

Small Bergman-Orlicz and Hardy-Orlicz spaces, and their composition operators

Charpentier

2019

Math. Z.

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We show that the weighted Bergman-Orlicz space A ψ α coincides with some weighted Banach space of holomorphic functions if and only if the Orlicz function ψ satisfies the so-called ∆ 2 -condition. In addition we prove that this condition characterizes those A ψ α on which every composition operator is bounded or order bounded into the Orlicz space L ψ α . This provides us with estimates of the norm and the essential norm of composition operators on such spaces. We also prove that when ψ satisfies the ∆ 2 -condition, a composition operator is compact on A ψ α if and only if it is order bounded into the so-called Morse-Transue space M ψ α . Our results stand in the unit ball of C N .

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Compact composition operators on Bergman-Orlicz spaces

Cited by 29 publications

References 19 publications

Estimates for approximation numbers of some classes of composition operators on the Hardy space

Estimates for approximation numbers of some classes of composition operators on the Hardy space

Some new properties of composition operators associated with lens maps

Small Bergman-Orlicz and Hardy-Orlicz spaces, and their composition operators

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