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M -ideals of compact operators into p

Kamil John, Dirk Werner (2000)

Czechoslovak Mathematical Journal

We show for 2 p < and subspaces X of quotients of L p with a 1 -unconditional finite-dimensional Schauder decomposition that K ( X , p ) is an M -ideal in L ( X , p ) .

M ( r , s ) -ideals of compact operators

Rainis Haller, Marje Johanson, Eve Oja (2012)

Czechoslovak Mathematical Journal

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 ( X , Y ) whenever 𝒦 ( X , X ) and 𝒦 ( Y , Y ) are M ( r 1 , s 1 ) - and M ( r 2 , s 2 ) -ideals in ( X , X ) and ( 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 ( X , Y ) is separably determined. Among others, our results complete and improve some well-known results...

Mapping Properties of c 0

Paul Lewis (1999)

Colloquium Mathematicae

Bessaga and Pełczyński showed that if c 0 embeds in the dual X * of a Banach space X, then 1 embeds as a complemented subspace of X. Pełczyński proved that every infinite-dimensional closed linear subspace of 1 contains a copy of 1 that is complemented in 1 . Later, Kadec and Pełczyński proved that every non-reflexive closed linear subspace of L 1 [ 0 , 1 ] contains a copy of 1 that is complemented in L 1 [ 0 , 1 ] . In this note a traditional sliding hump argument is used to establish a simple mapping property of c 0 which simultaneously...

Maps on idempotent operators

Peter Šemrl (2007)

Banach Center Publications

The set of all bounded linear idempotent operators on a Banach space X is a poset with the partial order defined by P ≤ Q if PQ = QP = P. Another natural relation on the set of idempotent operators is the orthogonality relation defined by P ⊥ Q ⇔ PQ = QP = 0. We briefly survey known theorems on maps on idempotents preserving order or orthogonality. We discuss some related results and open problems. The connections with physics, geometry, theory of automorphisms, and linear preserver problems will...

Markov convexity and local rigidity of distorted metrics

Manor Mendel, Assaf Naor (2013)

Journal of the European Mathematical Society

It is shown that a Banach space admits an equivalent norm whose modulus of uniform convexity has power-type p if and only if it is Markov p -convex. Counterexamples are constructed to natural questions related to isomorphic uniform convexity of metric spaces, showing in particular that tree metrics fail to have the dichotomy property.

Martingale operators and Hardy spaces generated by them

Ferenc Weisz (1995)

Studia Mathematica

Martingale Hardy spaces and BMO spaces generated by an operator T are investigated. An atomic decomposition of the space H p T is given if the operator T is predictable. We generalize the John-Nirenberg theorem, namely, we prove that the B M O q spaces generated by an operator T are all equivalent. The sharp operator is also considered and it is verified that the L p norm of the sharp operator is equivalent to the H p T norm. The interpolation spaces between the Hardy and BMO spaces are identified by the real method....

Matrix subspaces of L₁

Gideon Schechtman (2013)

Studia Mathematica

If E = e i and F = f i are two 1-unconditional basic sequences in L₁ with E r-concave and F p-convex, for some 1 ≤ r < p ≤ 2, then the space of matrices a i , j with norm | | a i , j | | E ( F ) = | | k | | l a k , l f l | | e k | | embeds into L₁. This generalizes a recent result of Prochno and Schütt.

Mazur spaces.

Wilansky, Albert (1981)

International Journal of Mathematics and Mathematical Sciences

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