We extend Troitsky’s ideas on measure-free martingales on Banach lattices to martingales of operators acting between a Banach lattice and a Banach space. We prove that each norm bounded martingale of cone absolutely summing (c.a.s.) operators (also known as 1-concave operators), from a Banach lattice E to a Banach space Y , can be generated by a single c.a.s. operator. As a consequence, we obta...

In this paper we introduce 2-JB*-triple-summing operators on real and complex JB*-triples. These operators generalize 2-C*-summing operators on C*-algebras. We also obtain a Pietsch’s factorization theorem in the setting of 2-JB*-triple-summing operators on JB*-triples.

Suppose that $${\mathscr {E}}$$ and {F}}$$ are Hilbert $$C^*$$ -modules. We present a power-norm $$\left( \left\| \cdot \right\| ^{{\mathscr {E}}}_n:n\in {\mathbb {N}}\right) $$ based on obtain some of its fundamental properties. introduce new definition the absolutely (2, 2)-summing operators from to , denote set such by $${\tilde{\Pi }}_2({\mathscr {E}},{\mathscr {F}})$$ with convention {E}})...

In 1973/74 Bennett and (independently) Carl proved that for 1 ≤ u ≤ 2 the identity map id: `u ↪→ `2 is absolutely (u, 1)-summing, i. e., for every unconditionally summable sequence (xn) in `u the scalar sequence (‖xn‖`2 ) is contained in `u, which improved upon well-known results of Littlewood and Orlicz. The following substantial extension is our main result: For a 2-concave symmetric Banach s...