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Denting point in the space of operator-valued continuous maps.

Ryszard Grzaslewicz, Samir B. Hadid (1996)

Revista Matemática de la Universidad Complutense de Madrid

In a former paper we describe the geometric properties of the space of continuous functions with values in the space of operators acting on a Hilbert space. In particular we show that dent B(L(H)) = ext B(L(H)) if dim H < 8 and card K < 8 and dent B(L(H)) = 0 if dim H < 8 or card K = 8, and x-ext C(K,L(H)) = ext C(K,L(H)).

Diameter-preserving maps on various classes of function spaces

Bruce A. Barnes, Ashoke K. Roy (2002)

Studia Mathematica

Under some mild assumptions, non-linear diameter-preserving bijections between (vector-valued) function spaces are characterized with the help of a well-known theorem of Ulam and Mazur. A necessary and sufficient condition for the existence of a diameter-preserving bijection between function spaces in the complex scalar case is derived, and a complete description of such maps is given in several important cases.

Dieudonné operators on the space of Bochner integrable functions

Marian Nowak (2011)

Banach Center Publications

A bounded linear operator between Banach spaces is called a Dieudonné operator ( = weakly completely continuous operator) if it maps weakly Cauchy sequences to weakly convergent sequences. Let (Ω,Σ,μ) be a finite measure space, and let X and Y be Banach spaces. We study Dieudonné operators T: L¹(X) → Y. Let i : L ( X ) L ¹ ( X ) stand for the canonical injection. We show that if X is almost reflexive and T: L¹(X) → Y is a Dieudonné operator, then T i : L ( X ) Y is a weakly compact operator. Moreover, we obtain that if T: L¹(X)...

Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces

Zoltán M. Balogh, Jeremy T. Tyson, Kevin Wildrick (2013)

Analysis and Geometry in Metric Spaces

We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincaré inequality. We show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space. For foliations of a metric space X defined by a David–Semmes regular mapping Π : X → W, we quantitatively estimate, in terms of Hausdorff dimension in W, the...

Duality of matrix-weighted Besov spaces

Svetlana Roudenko (2004)

Studia Mathematica

We determine the duals of the homogeneous matrix-weighted Besov spaces p α q ( W ) and p α q ( W ) which were previously defined in [5]. If W is a matrix A p weight, then the dual of p α q ( W ) can be identified with p ' - α q ' ( W - p ' / p ) and, similarly, [ p α q ( W ) ] * p ' - α q ' ( W - p ' / p ) . Moreover, for certain W which may not be in the A p class, the duals of p α q ( W ) and p α q ( W ) are determined and expressed in terms of the Besov spaces p ' - α q ' ( A Q - 1 ) and p ' - α q ' ( A Q - 1 ) , which we define in terms of reducing operators A Q Q associated with W. We also develop the basic theory of these reducing operator Besov spaces. Similar...

Duality on vector-valued weighted harmonic Bergman spaces

Salvador Pérez-Esteva (1996)

Studia Mathematica

We study the duals of the spaces A p α ( X ) of harmonic functions in the unit ball of n with values in a Banach space X, belonging to the Bochner L p space with weight ( 1 - | x | ) α , denoted by L p α ( X ) . For 0 < α < p-1 we construct continuous projections onto A p α ( X ) providing a decomposition L p α ( X ) = A p α ( X ) + M p α ( X ) . We discuss the conditions on p, α and X for which A p α ( X ) * = A q α ( X * ) and M p α ( X ) * = M q α ( X * ) , 1/p+1/q = 1. The last equality is equivalent to the Radon-Nikodým property of X*.

Duality theory of spaces of vector-valued continuous functions

Marian Nowak, Aleksandra Rzepka (2005)

Commentationes Mathematicae Universitatis Carolinae

Let X be a completely regular Hausdorff space, E a real normed space, and let C b ( X , E ) be the space of all bounded continuous E -valued functions on X . We develop the general duality theory of the space C b ( X , E ) endowed with locally solid topologies; in particular with the strict topologies β z ( X , E ) for z = σ , τ , t . As an application, we consider criteria for relative weak-star compactness in the spaces of vector measures M z ( X , E ' ) for z = σ , τ , t . It is shown that if a subset H of M z ( X , E ' ) is relatively σ ( M z ( X , E ' ) , C b ( X , E ) ) -compact, then the set conv ( S ( H ) ) is still relatively σ ( M z ( X , E ' ) , C b ( X , E ) ) -compact...

Dunford-Pettis operators on the space of Bochner integrable functions

Marian Nowak (2011)

Banach Center Publications

Let (Ω,Σ,μ) be a finite measure space and let X be a real Banach space. Let L Φ ( X ) be the Orlicz-Bochner space defined by a Young function Φ. We study the relationships between Dunford-Pettis operators T from L¹(X) to a Banach space Y and the compactness properties of the operators T restricted to L Φ ( X ) . In particular, it is shown that if X is a reflexive Banach space, then a bounded linear operator T:L¹(X) → Y is Dunford-Pettis if and only if T restricted to L ( X ) is ( τ ( L ( X ) , L ¹ ( X * ) ) , | | · | | Y ) -compact.

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