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}ddare mutually disjoint. A quasi-invertible operator is called aperiodic if for every n ∈ℕ and every non-zero band B ⊂ E, there exists a band A s.t. {0} ≠ A ⊂ B and A, {TA}dd , ... , {TnA}dd are mutually disjoint.
THEOREM. Let T : E → E be a quasi-invertible operator on a Dedekind complete Riesz space E. Then there exist T-reducing bands En (n ∈ ℕ ⋃ {∞}) such that the restriction of T to En (n ∈ ℕ) has strict period n, the restriction of T to E∞ is aperiodic and E = n∈ℕ ⋃ {∞}⊕ En.
Finally, the spectrum of bi-disjointness preserving operators is considered.
THEOREM. Let E be a Banach lattice which is either Dedekind complete or has a weak Fatou norm. Let T : E → E be a bi-disjointness preserving operator. If T is either of forward shift type, of backward shift type, hypernilpotent or aperiodic quasi-invertible, then the spectrum of T is rotationally invariant. If T is quasi-invertible with strict period n, then λ ∈ σ(T) implies λα ∈ σ(T) for any nth root of unity α.
The above theorems can be combined to deduce results concerning the spectrum of arbitrary bi-disjointness preserving operators. One such result is given below.
THEOREM. Let T : E → E be a bi-disjointness preserving operator on a Dedekind complete Banach lattice E. Suppose, for each 0 < r ∈ ℝ, {z ∈ ℂ : |z| = r} ⋂ σ(T) lies in an open half plane. Then there exists T-reducing bands E1 and E2 such that E = E1⊕ E2 , T|E1 is an abstract multiplication operator (i.e. is in the center of E) and T|E2 is quasi-nilpotent.
