Displaying similar documents to “The space of compact operators as an M-ideal in its bidual.”

Unconditional ideals of finite rank operators

Trond A. Abrahamsen, Asvald Lima, Vegard Lima (2008)

Czechoslovak Mathematical Journal

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Let X be a Banach space. We give characterizations of when ℱ ( Y , X ) is a u -ideal in 𝒲 ( Y , X ) for every Banach space Y in terms of nets of finite rank operators approximating weakly compact operators. Similar characterizations are given for the cases when ℱ ( X , Y ) is a u -ideal in 𝒲 ( X , Y ) for every Banach space Y , when ℱ ( Y , X ) is a u -ideal in 𝒲 ( Y , X * * ) for every Banach space Y , and when ℱ ( Y , X ) is a u -ideal in 𝒦 ( Y , X * * ) for every Banach space Y .

Unconditional ideals in Banach spaces

G. Godefroy, N. Kalton, P. Saphar (1993)

Studia Mathematica

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We show that a Banach space with separable dual can be renormed to satisfy hereditarily an “almost” optimal uniform smoothness condition. The optimal condition occurs when the canonical decomposition X * * * = X ⊥ ⊕ X * is unconditional. Motivated by this result, we define a subspace X of a Banach space Y to be an h-ideal (resp. a u-ideal) if there is an hermitian projection P (resp. a projection P with ∥I-2P∥ = 1) on Y* with kernel X ⊥ . We undertake a general study of h-ideals and u-ideals. For example...

M ( r , s ) -ideals of compact operators

Rainis Haller, Marje Johanson, Eve Oja (2012)

Czechoslovak Mathematical Journal

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We study the position of compact operators in the space of all continuous linear operators and its subspaces in terms of ideals. One of our main results states that for Banach spaces X and Y the subspace of all compact operators 𝒦 ( X , Y ) is an M ( r 1 r 2 , s 1 s 2 ) -ideal in the space of all continuous linear operators ℒ ( X , Y ) whenever 𝒦 ( X , X ) and 𝒦 ( Y , Y ) are M ( r 1 , s 1 ) - and M ( r 2 , s 2 ) -ideals in ℒ ( X , X ) and ℒ ( Y , Y ) , respectively, with r 1 + s 1 / 2 > 1 and r 2 + s 2 / 2 > 1 . We also prove that the M ( r , s ) -ideal 𝒦 ( X , Y ) in ℒ ( X , Y ) is separably determined. Among others, our results complete and improve some well-known...