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For two given symmetric sequence spaces E and F we study the (E,F)-multiplier space, that is, the space of all matrices M for which the Schur product M ∗ A maps E into F boundedly whenever A does. We obtain several results asserting continuous embedding of the (E,F)-multiplier space into the classical (p,q)-multiplier space (that is, when $E=l_p$, $F=l_q$). Furthermore, we present many examples of symmetric sequence spaces E and F whose projective and injective tensor products are not isomorphic to any subspace...

This paper studies the subset of singular traces generated by exponentiation-invariant extended limits. We describe relations between this subset and other important subsets of singular traces. We prove several conditions for measurability of operators from the weak trace class ideal with respect to the traces generated by exponentiation-invariant extended limits. We resolve an open question raised in [S. Lord, F. Sukochev, Measure theory in noncommutative spaces, SIGMA Symmetry Integrability Geom....

Let X, Y be Banach spaces and let (X,Y) be the space of bounded linear operators from X to Y. We develop the theory of double operator integrals on (X,Y) and apply this theory to obtain commutator estimates of the form $\left|\right|f\left(B\right)S-{Sf\left(A\right)\left|\right|}_{(X,Y)}\le const\left|\right|BS-{SA\left|\right|}_{(X,Y)}$ for a large class of functions f, where A ∈ (X), B ∈ (Y) are scalar type operators and S ∈ (X,Y). In particular, we establish this estimate for f(t): = |t| and for diagonalizable operators on $X={\ell }_p$ and $Y={\ell }_q$ for p < q. We also study the estimate above in the setting of Banach ideals...

Disjointification inequalities are proven for arbitrary martingale difference sequences and conditionally independent random variables of the form ${f_k\left(s\right)x_k\left(t\right)}_{k=1}ⁿ$, where $f_k$’s are independent and xk’s are arbitrary random variables from a symmetric space X on [0,1]. The main results show that the form of these inequalities depends on which side of L₂ the space X lies on. The disjointification inequalities obtained allow us to compare norms of sums of martingale differences and non-negative random variables with...