Citation
Hart, Dean Robert (1983) Disjointness Preserving Operators. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/88fv2p60. https://resolver.caltech.edu/CaltechTHESIS:03232017113645091
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 bidisjointness preserving if the order closure of TE 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 bidisjointness preserving. If E is in addition Dedekind complete, then the converse holds.
DEFINITION. Let T : E → E be a bidisjointness preserving operator. We say that T is:
(i) quasiinvertible if T is injective and {TE}^{dd} = E.
(ii) of forward 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 bidisjointness preserving operator on a Dedekind complete Riesz space E. Then there exist Treducing 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.
Quasiinvertible 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 quasiinvertible operator T has strict period n (n ∈ℕ) if T^{n} ∈ 0rth(E) and for every nonzero band B ⊂ E, there exists a band A s.t. {0} ≠ A ⊂ B and A, {TA}^{dd}, ... , {T^{n1}A}^{dd} are mutually disjoint. A quasiinvertible operator is called aperiodic if for every n ∈ℕ and every nonzero 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 quasiinvertible operator on a Dedekind complete Riesz space E. Then there exist Treducing 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 bidisjointness 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 bidisjointness preserving operator. If T is either of forward shift type, of backward shift type, hypernilpotent or aperiodic quasiinvertible, then the spectrum of T is rotationally invariant. If T is quasiinvertible 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 bidisjointness preserving operators. One such result is given below.
THEOREM. Let T : E → E be a bidisjointness 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 Treducing 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 quasinilpotent.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  Mathematics  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Mathematics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  20 May 1983  
Funders: 
 
Record Number:  CaltechTHESIS:03232017113645091  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:03232017113645091  
DOI:  10.7907/88fv2p60  
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:  14 Jun 2023 23:06 
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