scholarly journals Smoothness and norm attainment of bounded bilinear operators between Banach spaces

2019 ◽  
pp. 1-9
Author(s):  
Debmalya Sain
Keyword(s):  
Banach Spaces ◽  
Bilinear Operators

A general one-sided compactness result for interpolation of bilinear operators

2019 ◽  
Vol 124 (2) ◽  
pp. 247-262
Author(s):  
Eduardo Brandani Da Silva ◽  
Dicesar Lass Fernandez
Keyword(s):  
Banach Spaces ◽  
Interpolation Method ◽  
Compactness Theorem ◽  
Compactness Result ◽  
Interpolation Of Banach Spaces ◽  
Bilinear Operators

The behavior of bilinear operators acting on the interpolation of Banach spaces in relation to compactness is analyzed, and an one-sided compactness theorem is obtained for bilinear operators interpolated by the ρ interpolation method.

Interpolation of Bilinear Operators Between Quasi-Banach Spaces

Positivity ◽  
2006 ◽  
Vol 10 (3) ◽  
pp. 409-429 ◽  
Author(s):  
Loukas Grafakos ◽  
Mieczysław Mastyło
Keyword(s):  
Banach Spaces ◽  
Bilinear Operators

scholarly journals Some stability properties of Arens regular bilinear operators

1991 ◽  
Vol 34 (3) ◽  
pp. 443-454
Author(s):  
A. Ülger
Keyword(s):  
Banach Spaces ◽  
Compact Hausdorff Space ◽  
Borel Measure ◽  
Hausdorff Space ◽  
Bilinear Operator ◽  
Stability Properties ◽  
Bilinear Operators

In this paper we present three results about Arens regular bilinear operators. These are: (a). Let X, Y be two Banach spaces, K a compact Hausdorff space, µ a Borel measure on K and m: X × Y →ℂ a bounded bilinear operator. Then the bilinear operator defined by is regular iff m is regular, (b) Let (Xα), (Xα),(Zα) be three families of Banach spaces and let mα:Xα ×Yα→Zα, be a family of bilinear operators with supα∥mα∥<∞. Then the bilinear operator defined by is regular iff each mα, is regular, (c) Let X, Y have the Dieudonné property and let m:X × Y→Z be a bounded bilinear operator with m(X×Y) separable and such that, for each z′ in ext Z′1, z′∘m is regular. Then m is regular. Several applications of these results are also given.

scholarly journals On Bilinear Narrow Operators

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2892
Author(s):  
Marat Pliev ◽  
Nonna Dzhusoeva ◽  
Ruslan Kulaev
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Vector Lattice ◽  
Cartesian Product ◽  
Bilinear Operator ◽  
Projection Property ◽  
New Class ◽  
Bilinear Operators ◽  
Vector Lattices ◽  
Order Continuous

In this article, we introduce a new class of operators on the Cartesian product of vector lattices. We say that a bilinear operator T:E×F→W defined on the Cartesian product of vector lattices E and F and taking values in a vector lattice W is narrow if the partial operators Tx and Ty are narrow for all x∈E,y∈F. We prove that, for order-continuous Köthe–Banach spaces E and F and a Banach space X, the classes of narrow and weakly function narrow bilinear operators from E×F to X are coincident. Then, we prove that every order-to-norm continuous C-compact bilinear regular operator T is narrow. Finally, we show that a regular bilinear operator T from the Cartesian product E×F of vector lattices E and F with the principal projection property to an order continuous Banach lattice G is narrow if and only if |T| is.

scholarly journals Interpolation of compact bilinear operators among quasi-Banach spaces and applications

2018 ◽  
Vol 291 (14-15) ◽  
pp. 2168-2187 ◽  
Author(s):  
Fernando Cobos ◽  
Luz M. Fernández-Cabrera ◽  
Antón Martínez
Keyword(s):  
Banach Spaces ◽  
Bilinear Operators ◽  
Compact Bilinear Operators

scholarly journals Interpolation of compact bilinear operators

2020 ◽  
Vol 10 (02) ◽  
pp. 2050002
Author(s):  
Mieczysław Mastyło ◽  
Eduardo B. Silva
Keyword(s):  
Banach Spaces ◽  
Approximation Property ◽  
Interpolation Theorem ◽  
Real Interpolation ◽  
Bilinear Interpolation ◽  
Compactness Property ◽  
Interpolation Methods ◽  
Bilinear Operators ◽  
Compact Bilinear Operators ◽  
The Stability

This paper is devoted to the study of the stability of the compactness property of bilinear operators acting on the products of interpolated Banach spaces. We prove one-sided compactness results for bilinear operators on products of Banach spaces generated by abstract methods of interpolation, in the sense of Aronszajn and Gagliardo. To get these results, we prove a key one-sided bilinear interpolation theorem on compactness for bilinear operators on couples satisfying an extra approximation property. We give applications to general cases, including Peetre’s method and the general real interpolation methods.

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