A property of compact operators in the space of integrable functions

Бабенко, В. Ф.; Пичугов, С. А.

doi:10.1007/bf01085739

Cited by 22 publications

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“…Later, Babenko and Pichugov [1] showed that the same is true for a compact operator on Ll\ß, 1]. In general, if X is a Banach space and T G £{X) then T is said to satisfy Daugavet's equation if \\I -T\\ -1 + ||T|| [4].…”

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confidence: 99%

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A property of weakly compact operators on 𝐶[0,1]

Holub¹

1986

Proc. Amer. Math. Soc.

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confidence: 99%

“…This property has been found useful in work on a variety of problems in approximation theory (e.g. [4], and the references cited in [1]) and is of interest in the study of the metric geometry of Banach spaces of linear operators.…”

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A property of weakly compact operators on 𝐶[0,1]

Holub¹

1986

Proc. Amer. Math. Soc.

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“…In [2] Daugavet proved that if T is a compact operator on C{p), then ||/-r-T|| = 1 + ||r||, while Babenko and Pichugov [1] subsequently showed the same is true for a compact operator on L1{p) (see [6] for recent extensions). In general, if X is a Banach space and T G Z{X), then T is said to satisfy Daugavet's equation [4] if || J -T|| = 1 + |]T||.…”

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confidence: 99%

“…This property of an operator arises naturally in the consideration of problems of best approximation in function spaces where it has been utilized by a number of authors (e.g. [4], and the references cited in [1]). …”

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Daugavet’s equation and operators on 𝐿¹(𝜇)

Holub¹

1987

Proc. Amer. Math. Soc.

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ABSTRACT. Generalizing a result of Babenko and Pichugov, it is shown that if T is a weakly compact operator on Ll{p), where p is a rj-finite nonatomic measure, then \\L + T\\ = 1 + ||X||. A characterization of all operators T on L1^) having this property is also given.In [2] Daugavet proved that if T is a compact operator on C{p), then ||/-r-T|| = 1 + ||r||, while Babenko and Pichugov [1] subsequently showed the same is true for a compact operator on L1{p) (see [6] for recent extensions). In general, if X is a Banach space and T G Z{X), then T is said to satisfy Daugavet's equation In a recent paper [5] it was shown that Daugavet's equation actually holds for any weakly compact operator on C{p). The purpose of this note is to give an analogous extension of the theorem of Babenko and Pichugov to weakly compact operators by proving that if p is a rj-finite nonatomic measure on a space S, and T is a weakly compact operator on L1{p), then ||J + T|| = 1 + ||T|| (Theorem 1). Related results concerning extensions and generalizations of this theorem, including a characterization of all operators Ton L1{p) for which ||J-r-T|| = 1+||T|| (Theorem 2) are also given.We begin with a simple result concerning the norming of operators on L1{p). As usual we denote by Z{X) the space of all bounded linear operators on a Banach space A, by \A the characteristic function of a measurable set A c S, and by ||/||i and 11g||oo) the norms of functions / G Ll{p) and g G L°°{p), respectively. The complement of a set A in 5 is denoted by A'. PROPOSITION 1. LetT G ¿{L1^)).Then given any e > 0 there is a measurable set Ac S for which 0 < m{A) < e and ||T(xA/m(A))||i > ||T|| -e.PROOF. Since p is tr-finite, {Ll{p))* = L°°{p). Therefore if T G t{Ll{p)), then T* G £{L°°{p)) and ||r|| = ||T*|| = sup,|B,|oo = 1 ||T*ff||oo. Hence given any e > 0 there is a function g G L°°{p) for which ¡¡pH«, = 1 and \\T*g\\oo > \\T\\-e/2. Since ||T*g||oo = ess supt \{T*g){t)\ it follows that there is a measurable set A c S for which m{A) > 0 and \{T*g){t)\ > \\T\\ -s for all t G S. Replacing A (if necessary) by a subset B c A for which 0 < m{B) < e and on which the sign of {T*g){t) is constant (which may be done since p is nonatomic), and replacing g by (-g) if

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“…Originally, (1) was established by Daugavet for compact operators on C[0, 1] (see [2]). The case of L 1 was first treated by Lozanovskii in his paper [6], where he proved Daugavet's theorem for compact operators in L 1 [0, 1] (see also [1]). Later, Holub generalized this result for all weakly compact operators on an arbitrary atomless L 1 (Ω) (see [4]).…”

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confidence: 99%