Disjointness Preserving Operators

Citation

Hart, Dean Robert (1983) Disjointness Preserving Operators. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/88fv-2p60. https://resolver.caltech.edu/CaltechTHESIS:03232017-113645091

Abstract

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 E_i (i = 1,2,3,4) such that _i=1⊕⁴ E_i = E and the restriction of T to E_i satisfies the ith property listed in the preceding definition.

Quasi-invertible operators can be decomposed further in the following way. Set 0rth(E) :={T ∈ ℒ_b(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, ... , {T^n-1A}^dd are 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 E_n (n ∈ ℕ ⋃ {∞}) such that the restriction of T to E_n (n ∈ ℕ) has strict period n, the restriction of T to E_∞ is aperiodic and E = _{n∈ℕ ⋃ {∞}}⊕ E_n.

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.

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