The strong Gelfand–Phillips property in Banach lattices

Abstract

We introduce the concept of the strong Gelfand-Phillips (GP) property in Banach lattices, and we characterize Banach lattices with the strong GP property. Next, by introducing the class of almost limited completely continuous operators from an arbitrary Banach lattice E to another F , we give some properties of them related to some well-known classes of operators and, especially, to the strong GP property of the Banach lattice E.

On the Schur, positive Schur and weak Dunford-Pettis properties in Fréchet lattices

“…A Banach lattice has the positive Schur property if weakly null sequences formed by positive vectors are norm null. A lot of research has been done on this property, for some recent contributions see, e.g., [6,7,13,18,25,26,27,29]. Among other results, in this paper we prove that, contrary to the case of the Schur property for Banach spaces, the Schur and the positive Schur properties are 3-lattice properties.…”

On the Schur, positive Schur and weak Dunford–Pettis properties in Fréchet lattices

“…A property P of Banach lattices is a 3-lattice property if the following holds: given a closed ideal I of the Banach lattice E, if two of the lattices E, I and E/I have P, then the third one has P as well.A Banach lattice has the positive Schur property if weakly null sequences formed by positive vectors are norm null. A lot of research has been done on this property, for some recent contributions see, e.g., [6,7,13,18,24,25,26,28]. Among other results, in this paper we prove that, contrary to the case of the Schur property for Banach spaces, the Schur and the positive Schur properties are 3-lattice properties.…”

“…A Banach lattice has the positive Schur property if weakly null sequences formed by positive vectors are norm null. A lot of research has been done on this property, for some recent contributions see, e.g., [6,7,13,18,24,25,26,28]. Among other results, in this paper we prove that, contrary to the case of the Schur property for Banach spaces, the Schur and the positive Schur properties are 3-lattice properties.…”

The positive polynomial Schur property in Banach lattices

“…In the setting of Banach lattices, the positive Schur property (weakly null sequences formed by positive vectors are norm null) has been extensively studied, recent developments can be found in [5,8,10,16,33,36,37]. So it is a natural step to consider the lattice counterpart of the polynomial Schur property, which is the main subject of this paper.…”