Invertible weighted composition operators on weighted function spaces

Singh, Raj; Manhas, Jasbir Singh

doi:10.1007/bf01904058

Cited by 17 publications

(17 citation statements)

References 12 publications

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“…If it is well defined, then, by the closed graph theorem, it is a bounded linear operator, and consequently µ • φ −1 µ and h k := dµ • φ −k /dµ ∈ L ∞ (µ) for every integer k ≥ 0 (see [25] for more details). THEOREM 3.1.…”

Section: Composition Operatorsmentioning

confidence: 99%

L(n)-HYPONORMALITY: A MISSING BRIDGE BETWEEN SUBNORMALITY AND PARANORMALITY

Jung¹,

Park²,

Stochel³

2010

J. Aust. Math. Soc.

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A new notion of L(n)-hyponormality is introduced in order to provide a bridge between subnormality and paranormality, two concepts which have received considerable attention from operator theorists since the 1950s. Criteria for L(n)-hyponormality are given. Relationships to other notions of hyponormality are discussed in the context of weighted shift and composition operators.2000 Mathematics subject classification: primary 47B20; secondary 47B33, 47B37.

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Section: Composition Operatorsmentioning

confidence: 99%

L(n)-HYPONORMALITY: A MISSING BRIDGE BETWEEN SUBNORMALITY AND PARANORMALITY

Jung¹,

Park²,

Stochel³

2010

J. Aust. Math. Soc.

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“…The interested reader is referred to [10], [28] and [38] for further information on composition operators.…”

Section: Preliminariesmentioning

confidence: 99%

Joint subnormality of n-tuples and C₀-semigroups of composition operators on L²-spaces, II

Budzyński¹,

Stochel²

2009

Studia Math.

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Abstract. In the previous paper, we have characterized (joint) subnormality of a C0-semigroup of composition operators on L 2 -space by positive definiteness of the RadonNikodym derivatives attached to it at each rational point. In the present paper, we show that in the case of C0-groups of composition operators on L 2 -space the positive definiteness requirement can be replaced by a kind of consistency condition which seems to be simpler to work with. It turns out that the consistency condition also characterizes subnormality of C0-semigroups of composition operators on L 2 -space induced by injective and bimeasurable transformations. The consistency condition, when formulated in the language of the Laplace transform, takes a multiplicative form. The paper concludes with some examples. [21] proved that a C 0 -semigroup of subnormal operators is automatically jointly subnormal and as such has an extension to a C 0 -semigroup of normal operators. As noted below, this is still true for C 0 -groups of subnormal operators (cf. Proposition 2.1). Itô's theorem enabled Nussbaum [29] to prove the subnormality of the infinitesimal generator of a C 0 -semigroup of subnormal operators. The interested reader is referred to the monograph [7] for the foundations of the theory of subnormal operators.Composition operators, which play an essential role in ergodic theory, turn out to be interesting objects of operator theory. The questions of bound-

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“…The function ϕ induces the composition operator C ϕ , defined on the space of holomorphic functions on U by C ϕ f = f •ϕ. The restriction of C ϕ to various Banach spaces of holomorphic functions on U has been an active subject of research for more than three decades, and it will continue to be for decades to come (see [15], [16] and [5]). Let D denote the Dirichlet space of analytic functions on the unit disk with derivatives that are square integrable with respect to the area measure on the disk.…”

Section: Introductionmentioning

confidence: 99%

Self-commutators of automorphic composition operators on the Dirichlet space

Abdollahi

2008

Proc. Amer. Math. Soc.

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Abstract. Let ϕ be a conformal automorphism on the unit disk U and C ϕ : D −→ D be the composition operator on the Dirichlet space D induced by ϕ. In this article we completely determine the point spectrum, spectrum, essential spectrum and essential norm of the operators C * ϕ C ϕ , C ϕ C * ϕ and self-commutators of C ϕ , which expose that the spectrum and point spectrum coincide. We also find the eigenfunctions of the operators.

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Invertible weighted composition operators on weighted function spaces

Cited by 17 publications

References 12 publications

L(n)-HYPONORMALITY: A MISSING BRIDGE BETWEEN SUBNORMALITY AND PARANORMALITY

L(n)-HYPONORMALITY: A MISSING BRIDGE BETWEEN SUBNORMALITY AND PARANORMALITY

Joint subnormality of n-tuples and C₀-semigroups of composition operators on L²-spaces, II

Self-commutators of automorphic composition operators on the Dirichlet space

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Invertible weighted composition operators on weighted function spaces

Cited by 17 publications

References 12 publications

L(n)-HYPONORMALITY: A MISSING BRIDGE BETWEEN SUBNORMALITY AND PARANORMALITY

L(n)-HYPONORMALITY: A MISSING BRIDGE BETWEEN SUBNORMALITY AND PARANORMALITY

Joint subnormality of n-tuples and C0-semigroups of composition operators on L2-spaces, II

Self-commutators of automorphic composition operators on the Dirichlet space

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Joint subnormality of n-tuples and C₀-semigroups of composition operators on L²-spaces, II