Strong Factorizations between Couples of Operators on Banach Function Spaces

Let

and

S\colon X_2\to Y_2 S  ⁣ : X 2 → Y 2

be two continuous linear operators between Banach function spaces related to a finite measure space. Under some lattice requirements on the spaces involved, we give characterizations by means of inequalities of when

T T

can be strongly factorized through

S S

, that is,

T=M_g\circ S\circ M_f T = M g ∘ S ∘ M f

with

M_f\colon X_1\to X_2 M f  ⁣ : X 1 → X 2

and

M_g\colon Y_2\to Y_1 M g  ⁣ : Y 2 → Y 1

being multiplication operators defined by some measurable functions

f f

and

g g

. In particular, we study the cases when

S S

is a composition operator or a kernel operator.