Disjointness Preserving Operators

Let E and F be Archimedian Riesz spaces. A linear operator T : E → F is called disjointness preserving if |f| ∧ |g| = 0 in E implies |Tf| ∧ |Tg| = 0 in F. An order continuous disjointness preserving operator T : E → E is called bi-disjointness preserving if the order closure of |T|E is an ideal in E. If the order dual of E separates the points of E, then every order continuous disjointness preserving operator whose adjoint is disjointness preserving is bi-disjointness preserving. If E is in addition Dedekind complete, then the converse holds. DEFINITION. Let T : E → E be a bi-disjointness preserving operator. We say that T is: (i) quasi-invertible if T is injective and {TE}dd = E. (ii) of forward shift type if T is injective and n=1∩∞{TnE}dd = {0}. (iii) of backward shift type if n=1∨∞ Ker Tn = E and{TE}dd = E. (iv) hypernilpotent if n=1∨∞ Ker Tn = E and n=1∩∞ {TnE}dd = {0}. The supremum in (iii) and (iv) is taken in the Boolean algebra of bands. The following decomposition theorem is proved. THEOREM. Let T : E → E be a bi-disjointness preserving operator on a Dedekind complete Riesz space E. Then there exist T-reducing bands Ei (i = 1,2,3,4) such that i=1⊕4 Ei = E and the restriction of T to Ei satisfies the ith property listed in the preceding definition. Quasi-invertible operators can be decomposed further in the following way. Set 0rth(E) :={T ∈ ℒb(E) : TB ⊂ B for every band B}. We say that a quasi-invertible operator T has strict period n (n ∈ℕ) if Tn ∈ 0rth(E) and for every non-zero band B ⊂ E, there exists a band A s.t. {0} ≠ A ⊂ B and A, {TA}dd, ... , {Tn-1A}dd are mutually disjoint. A quasi-invertible operator i