TAILIEUCHUNG - Alternative Polynomial and Holomorphic Dunford-Pettis properties

Alternatives to the Polynomial Dunford-Pettis property and the Holomorphic Dunford-Pettis property, called the PDP1 and HDP1 properties, respectively, are introduced. These are shown to be equivalent to the DP1 property, an alternative Dunford-Pettis property previously introduced by the author, thus mirroring the equivalence of the three original properties. | Turk J Math 23 (1999) , 407 – 415. ¨ ITAK ˙ c TUB ALTERNATIVE POLYNOMIAL AND HOLOMORPHIC DUNFORD-PETTIS PROPERTIES ∗ Walden Freedman Abstract Alternatives to the Polynomial Dunford-Pettis property and the Holomorphic Dunford-Pettis property, called the PDP1 and HDP1 properties, respectively, are introduced. These are shown to be equivalent to the DP1 property, an alternative Dunford-Pettis property previously introduced by the author, thus mirroring the equivalence of the three original properties. Introduction In [4], R. Ryan proved that the Dunford-Pettis property, the Polynomial DunfordPettis property, and the Holomorphic Dunford-Pettis property are all equivalent. In [1], a property closely related to the Dunford-Pettis property, called the DP1 property, is introduced and defined as follows: A Banach space X has the DP1 property if for any Banach space Y and any weakly compact linear operator T : X → Y , if xn → x weakly in X with kxn k = kxk = 1 for all n, then T xn → T x in norm in Y . We will consider two alternative properties, the PDP1 property and the HDP1 property, in the same spirit as [4], and show that like the original properties, DP1, PDP1 and HDP1 are all equivalent. Some applications to Banach algebras are also given. Notation and Background Throughout the paper, X and Y will denote Banach spaces over the field of complex numbers. We identify X with its image in X ∗∗ under its canonical embedding in X ∗∗ . The Banach space of all bounded linear operators from X to Y will be denoted L(X; Y ). Given x0 ∈ X and r > 0, the open and closed balls centered at x0 with radius r will be denoted ∆(x0 , r) and ∆(x0 , r), respectively. By the term ‘operator’, we will always ∗ This paper is a part of the author’s doctoral dissertation at the University of California, Santa Barbara. 407 FREEDMAN mean a bounded linear operator. Given f ∈ X and a ∈ X ∗ we often write f(a) or hf, ai for the evaluation of a on f. The Banach space of all sequences in Y that .

• Toán học
• Alternative Polynomial
• Holomorphic Dunford
• Pettis property
• HDP1 properties
• DP1 property
• Original properties
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