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BanachJ.Math.Anal.6(2012),no.1,61–73 OPTIMALRANGETHEOREMSFOROPERATORSWITH B J M A p -THPOWERFACTORABLEADJOINTS

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Banach J. Math. Anal. 6 (2012), no. 1, 61–73

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anach

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ournal of

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athematical

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nalysis ISSN: 1735-8787 (electronic)

www.emis.de/journals/BJMA/

OPTIMAL RANGE THEOREMS FOR OPERATORS WITH p-TH POWER FACTORABLE ADJOINTS

ORLANDO GALDAMES BRAVO1 AND ENRIQUE A. S ´ANCHEZ P ´EREZ2∗ Communicated by M. Abel

Abstract. Consider an operator T : E →X(µ) from a Banach space E to a Banach function spaceX(µ) over a finite measureµ such that its dual map isp-th power factorable. We compute the optimal range ofT that is defined to be the smallest Banach function space such that the range of T lies in it and the restricted operator has p-th power factorable adjoint. For the case p= 1, the requirement onT is just continuity, so our results give in this case the optimal range for a continuous operator. We give examples from classical and harmonic analysis, as convolution operators, Hardy type operators and the Volterra operator.

1Instituto Universitario de Matem´atica Pura y Aplicada, Universidad Polit´ecnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain.

E-mail address: [email protected]

2Instituto Universitario de Matem´atica Pura y Aplicada, Universidad Polit´ecnica de Valencia Camino de Vera s/n, 46022 Valencia, Spain.

E-mail address: [email protected]

Date: Received: 26 August 2011; Accepted: 9 September 2011.

∗ Corresponding author.

2010Mathematics Subject Classification. Primary 47B38, Secondary 46E30.

Key words and phrases. Banach function space, operator, vector measure, integration, opti- mal range.

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