Abstract Spectral Decompositions Guaranteed by the Hilbert Transform

Berkson, Earl; Gillespie, T. A.; Muhly, Paul S.

doi:10.1112/plms/s3-53.3.489

Cited by 70 publications

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“…Berkson, Gillespie and Muhly have shown by transference that A is associated with a resolution of the identity with respect to an increasing family E(s) (0 ≤ s < ∞) of uniformly bounded projections in B(X) that is strongly right-continuous and has strong left-hand limits; see [4]. Further there is an integral with respect to this spectral family, and a bounded functional calculus map which we now describe.…”

Section: Examples and Bounded H ∞ Functional Calculusmentioning

confidence: 89%

A maximal theorem for holomorphic semigroups

Blower

Doust

2005

The Quarterly Journal of Mathematics

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Let X be a closed linear subspace of the Lebesgue space L p (Ω, µ) for some 1 < p < ∞, and let −A be an invertible operator that is the generator of a bounded holomorphic semigroup T t on X. Then for each 0 < α < 1 the maximal function sup t>0 |T t f (x)| belongs to L p (Ω, µ) for each f in the domain of A α . If moreover iA generates a bounded C 0 -group and A has spectrum contained in (0, ∞), then A has a bounded H ∞ functional calculus.

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Section: Examples and Bounded H ∞ Functional Calculusmentioning

confidence: 89%

A maximal theorem for holomorphic semigroups

Blower

Doust

2005

The Quarterly Journal of Mathematics

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“…In Section 2, the singular series operators are defined as convolution operators on the sequence spaces p , 1 ≤ p < ∞, and we show that the associated maximal operator is bounded on p for 1 < p < ∞ and is of weak type (1,1). This result can be proved by standard real variable methods using Calderón-Zygmund decomposition.…”

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confidence: 86%

“…, the kernels {φ N } satisfy the hypothesis of Corollary 2.4.5 of [10], so that the operator T φ (T ∧ φ in the notation of [10]) is bounded on p for 1 < p < ∞ and is of weak type (1,1).…”

Section: With Proposition 22 and (S3)mentioning

confidence: 99%

On ergodic singular integral operators

Alphonse¹,

Madan²

1993

Colloq. Math.

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1. Introduction. In [11] Petersen gave a direct proof of Cotlar's [8] result on the existence a.e. and boundedness of the ergodic Hilbert transform defined by a measure-preserving invertible transformation on a probability space (X, µ). Petersen's proof consists in proving p -inequalities for the maximal discrete Hilbert transform on sequence spaces and then applying Calderón's transference principle ([6], also [7]).In this paper we study a class of operators, called singular series operators (Definition 2.1), which are discrete analogues of singular integral operators on R ([13], [14]). By transference, we then consider the corresponding ergodic operators on L p -spaces of Banach space valued functions on X, for suitable Banach spaces B.In Section 2, the singular series operators are defined as convolution operators on the sequence spaces p , 1 ≤ p < ∞, and we show that the associated maximal operator is bounded on p for 1 < p < ∞ and is of weak type (1, 1). This result can be proved by standard real variable methods using Calderón-Zygmund decomposition. We prove the maximal operator inequalities by transferring these from the corresponding inequalities on R. . For a geometric characterization, we refer to [5].In Section 3, we define the ergodic singular operators and using the transference principle as in [11], we prove the existence a.e. and boundedness of these operators on the spaces L p B (X) consisting of B-valued (strongly)

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“…Other examples include the Schatten p-classes for 1 < p < ∞ (Gutiérrez [22], Bourgain [6]), and the noncommutative L p (M, τ )-spaces associated with a von Neumann algebra M , equipped with a normal, semifinite, faithful (abbreviated as n.s.f.) trace τ (Berkson, Gillespie and Muhly [3]). We refer to Burkholder [10,12] for more properties of U M D Banach spaces, connections to other topics and further references.…”

Section: Introductionmentioning

confidence: 99%

On the operator space $UMD$ property for noncommutative $L_p$-spaces

Musat¹

2006

Indiana Univ. Math. J.

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Abstract. We study the operator space U M D property, introduced by Pisier in the context of noncommutative vector-valued Lp-spaces. It is unknown whether the property is independent of p in this setting. We prove that for 1 < p, q < ∞ , the Schatten q-classes Sq are OU M Dp . The proof relies on properties of the Haagerup tensor product and complex interpolation. Using ultraproduct techniques, we extend this result to a large class of noncommutative Lq-spaces. Namely, we show that if M is a QW EP von Neumann algebra (i.e., a quotient of a C * -algebra with Lance's weak expectation property) equipped with a normal, faithful tracial state τ , then Lq(M, τ ) is OU M Dp for 1 < p, q < ∞ .

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Abstract Spectral Decompositions Guaranteed by the Hilbert Transform

Cited by 70 publications

References 0 publications

A maximal theorem for holomorphic semigroups

A maximal theorem for holomorphic semigroups

On ergodic singular integral operators

On the operator space $UMD$ property for noncommutative $L_p$-spaces

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