Strictly singular and power-compact operators on Banach lattices (Q1760336)

An operator between Banach spaces is strictly singular if none of its restrictions to an infinite dimensional subspace is an isomorphic embedding. Strictly singular operators form a closed operator ideal containing the compact operators. ``Measuring'' the gap between strictly singular and compact operators on a Banach space, intensified in the last 20 years, plays an important role in understanding the structure of the space. The authors study possible extensions of a classical result of \textit{V. D. Mil'man} [Teor. Funkts., Funkts. Anal. Prilozh. 10, 15--26 (1970; Zbl 0217.17201)], stating that any strictly singular operator on an \(L_p\)-space has a compact square. They consider the class of disjointly homogeneous Banach lattices, containing \(L_p\)-spaces, Lorentz function spaces, certain Orlicz function spaces, as well as Tsirelson-type spaces. A~Banach lattice is called disjointly homogeneous if every pair of its disjoint normalized sequences admit equivalent subsequences. If any disjoint normalized sequence has a subsequence equivalent to the unit vector basis of \(\ell_p\), then the lattice is called \(p\)-disjointly homogeneous. The authors present several positive results, in particular they prove that any strictly singular operator on a disjointly homogeneous Banach lattice with finite cotype and unconditional basis has a compact square, and provide certain mild conditions on a \(p\)-disjointly homogeneous lattice ensuring that any strictly singular operator on it has a compact square or is compact itself. The paper also contains some interesting examples of Banach lattices admitting strictly singular operators with no compact power, including certain Orlicz spaces and \(\ell_p\)-sums of \(p\)-convexified Tsirelson spaces.