Closed ideals of operators acting on some families of sequence spaces

Abstract

We study the lattice of closed ideals in the algebra of continuous linear operators acting on pth Tandori and p ′ th Cesàro sequence spaces, 1 p < ∞, which we show are isomorphic to the classical sequence spaces (⊕ ∞ n=1 ℓ n ∞ ) p and (⊕ ∞ n=1 ℓ n 1 ) p ′ , respectively. We also show that Tandori sequence spaces are complemented in certain Lorentz sequence spaces, and that the lattice of closed ideals for certain other Lorentz and Garling sequence spaces has infinite cardinality.2010 Mathematics Subject Classification. Primary 46B45; Secondary 46B25, 47A99.

Local approach to order continuity in Cesàro function spaces

“…with n ∈ N is the discrete Hardy operator. Over the last two decades, research on the structure of the above-mentioned spaces (which we will collectively call spaces related to decreasing functions) has enjoyed unwavering interest from many different viewpoints like, for example, [Sin01] and [Sin03]; • operator ideals [Wal20] (see also survey papers [AM14], [FLM16] and [Sin07]). For more information, we also refer to two monographs [Ben96] and [G-E98].…”

A new complemented subspace for the Lorentz sequence spaces, with an application to its lattice of closed ideals

“…In [KPSTT12], it was shown (among other results) that the lattice of closed ideals for the operator algebra can be put into a chain: Here, denotes the compact operators, the strictly singular operators, and the ideal of operators which fail to be bounded below on any isomorph of . While, in [Wa20, Corollary 2.7], for the special case where and , a chain of distinct closed ideals with cardinality of the continuum were identified lying between and , for the general case, the only distinct elements known were those of the above chain.…”

“…Until recently, the only nontrivial complemented subspace discussed in the literature was [ACL73]. Then, in [Wa20], it was shown that for certain weights (see Theorem 2.2 below), the space contains a 1-complemented subspace isomorphic to . Up to now, these were the only nontrivial complemented subspaces known to exist.…”

A new complemented subspace for the Lorentz sequence spaces, with an application to its lattice of closed ideals

“…Until recently, the only nontrivial complemented subspace discussed in the literature was ℓ p ( [ACL73]). Then, in [Wa20] it was shown that for certain weights w (see Theorem 2.2 below), the space d(w, p) contains a 1complemented subspace isomorphic to ( ∞ n=1 ℓ n ∞ ) p . Up to now, these were the only nontrivial complemented subspaces known to exist.…”