let $t$ be a bounded operator on the banach space $x$ and $ph$ be an analytic self-map of the unit disk $bbb{d}$‎. ‎we investigate some operator theoretic properties of‎ ‎bilateral composition operator $c_{ph‎, ‎t}‎: ‎f ri t circ f circ ph$ on the vector-valued hardy space $h^p(x)$ for $1 leq p leq‎ ‎+infty$.‎ ‎compactness and weak compactness of $c_{ph‎, ‎t}$ on $h^p(x)$‎ ‎are characterized an...

In this paper we prove L estimates for a tri-linear operator, whose symbol is given by the product of two standard symbols, satisfying the well knownMarcinkiewiczHörmander-Mihlin condition. Our main result contains in particular the classical CoifmanMeyer theorem. This tri-linear operator is the simplest example of a large class of multilinear operators, which we called paraproducts with flag s...

We give a systematic study on the Hardy spaces of functions with values in the non-commutative L-spaces associated with a semifinite von Neumann algebra M. This is motivated by the works on matrix valued Harmonic Analysis (operator weighted norm inequalities, operator Hilbert transform), and on the other hand, by the recent development on the non-commutative martingale inequalities. Our non-com...