## Citation

Hart, Dean Robert
(1983)
*Disjointness Preserving Operators.*
Dissertation (Ph.D.), California Institute of Technology.
http://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 foruard shift type if T is injective and _{n=1}∩^{∞}{T^{n}E}^{dd} = {0}.

(iii) of backward shift type if _{n=1}∨^{∞} Ker T^{n} = E and{TE}^{dd} = E.

(iv) hypernilpotent if _{n=1}∨^{∞} Ker T^{n} = E and _{n=1}∩^{∞} {T^{n}E}^{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 T^{n} ∈ 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-1}A}^{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} , ... , {T^{n}A}^{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 E_{1} and E_{2} such that E = E_{1}⊕ E_{2} , T|_{E1} is an abstract multiplication operator (i.e. is in the center of E) and T|_{E2} is quasi-nilpotent.

Item Type: | Thesis (Dissertation (Ph.D.)) | ||||
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Subject Keywords: | Mathematics | ||||

Degree Grantor: | California Institute of Technology | ||||

Division: | Physics, Mathematics and Astronomy | ||||

Major Option: | Mathematics | ||||

Thesis Availability: | Restricted to Caltech community only | ||||

Research Advisor(s): | - Luxemburg, W. A. J.
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Thesis Committee: | - Unknown, Unknown
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Defense Date: | 20 May 1983 | ||||

Funders: |
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Record Number: | CaltechTHESIS:03232017-113645091 | ||||

Persistent URL: | http://resolver.caltech.edu/CaltechTHESIS:03232017-113645091 | ||||

Default Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||

ID Code: | 10101 | ||||

Collection: | CaltechTHESIS | ||||

Deposited By: | Bianca Rios | ||||

Deposited On: | 24 Mar 2017 14:42 | ||||

Last Modified: | 24 Mar 2017 14:42 |

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