Remarks on Banach spaces of compact operators

Johnson, Jerry

doi:10.1016/0022-1236(79)90042-9

Cited by 49 publications

(31 citation statements)

References 16 publications

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“…We shall need the following well-known result of J. Johnson [22]. Let (K α ) be a shrinking M(C)AI for X.…”

Section: Lemma 42 Let X Be a Banach Space And T ∈ B L(x)mentioning

confidence: 99%

Geometry of Banach spaces having shrinking approximations of the identity

Oja

2000

Trans. Amer. Math. Soc.

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Abstract. Let a, c ≥ 0 and let B be a compact set of scalars. We introduce property M * (a, B, c) of Banach spaces X by the requirement that lim supwhenever (x * ν ) is a bounded net converging weak * to x * in X * and y * ≤ x * . Using M * (a, B, c) with max |B| + c > 1, we characterize the existence of certain shrinking approximations of the identity (in particular, those related to M -, u-, and h-ideals of compact or approximable operators). We also show that the existence of these approximations of the identity is separably determined.

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“…We shall need the following well-known result of J. Johnson [22]. Let (K α ) be a shrinking M(C)AI for X.…”

Section: Lemma 42 Let X Be a Banach Space And T ∈ B L(x)mentioning

confidence: 99%

Geometry of Banach spaces having shrinking approximations of the identity

Oja

2000

Trans. Amer. Math. Soc.

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“…Asvald Lima and Eve Oj a [12] is linear and therefore it is a Hahn-Banach extension operator. Thus </> z .…”

Section: X)) F Ex(z\y)\mentioning

confidence: 99%

Ideals of compact operators

Lima¹,

Oja²

2004

J. Aust. Math. Soc.

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We give an example of a Banach space X such that J^(X, X) is not an ideal in J(f(X, X**). We prove that if z* is a weak* denting point in the unit ball of Z* and if X is a closed subspace of a Banach space Y, then the set of norm-preserving extensions HB(x*®z*) c jSf(Z*, Y)* of a functional x*®z* e (Z®X)* is equal to the set HB(x*) ® (z*}. Using this result, we show that if X is an W-ideal in Y and Z is a reflexive Banach space, then Jf(Z, X) is an W-ideal in Jf(Z, Y) whenever JT(Z, X) is an ideal in Jf(Z, Y).We also show that X ( Z , X) is an ideal (respectively, an A/-ideal) in Jf(Z, Y) for all Banach spaces Z whenever X is an ideal (respectively, an M-ideal) in Y and X* has the compact approximation property with conjugate operators.2000 Mathematics subject classification: primary 46B20,46B28, 47L05.

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“…with Φ λ can be constructed more or less explicitly (as it was by Johnson in the proof of Lemma 1 in [10] …”

Section: 52mentioning

confidence: 99%

Operators That Are Nuclear Whenever They Are Nuclear for a Larger Range Space

Oja¹

2004

Proceedings of the Edinburgh Mathematical Society

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Let X be a Banach space and let Y be a closed subspace of a Banach space Z. The following theorem is proved. Assume that X * or Z * has the approximation property. If there exists a bounded linear extension operator from Y * to Z * , then any bounded linear operator T : X → Y is nuclear whenever T is nuclear from X to Z. The particular case of the theorem with Z = Y * * is due to Grothendieck and Oja and Reinov. Numerous applications are presented. For instance, it is shown that a bounded linear operator T from an arbitrary Banach space X to an L∞-space Y is nuclear whenever T is nuclear from X to some Banach space Z containing Y as a subspace.

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Remarks on Banach spaces of compact operators

Cited by 49 publications

References 16 publications

Geometry of Banach spaces having shrinking approximations of the identity

Geometry of Banach spaces having shrinking approximations of the identity

Ideals of compact operators

Operators That Are Nuclear Whenever They Are Nuclear for a Larger Range Space

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