A Note on Disjointness Preserving Operators

1984 ◽  
Vol 90 (4) ◽  
pp. 543 ◽  
Author(s):  
B. De Pagter
Keyword(s):  
Disjointness Preserving

scholarly journals Correction to: Disjointness preserving operators on normed pre-Riesz spaces: extensions and inverses

Positivity ◽  
2020 ◽  
Vol 24 (2) ◽  
pp. 505-505
Author(s):  
Anke Kalauch ◽  
Onno van Gaans ◽  
Feng Zhang
Keyword(s):  
Riesz Spaces ◽  
Disjointness Preserving

scholarly journals A solution to a problem on invertible disjointness preserving operators

1998 ◽  
Vol 126 (5) ◽  
pp. 1501-1505 ◽  
Author(s):  
Yuri A. Abramovich ◽  
Arkady K. Kitover
Keyword(s):  
Disjointness Preserving

scholarly journals Unbounded Disjointness Preserving Linear Functionals

2004 ◽  
Vol 141 (1) ◽  
pp. 21-32 ◽  
Author(s):  
Lawrence G. Brown ◽  
Ngai-Ching Wong
Keyword(s):  
Linear Functionals ◽  
Disjointness Preserving

Bijective Disjointness Preserving Operators

1998 ◽  
pp. 1-8
Author(s):  
Yuri Abramovich ◽  
Arkady Kitover
Keyword(s):  
Disjointness Preserving

Sums of Order Bounded Disjointness Preserving Linear Operators

2019 ◽  
Vol 60 (1) ◽  
pp. 114-123
Author(s):  
A. G. Kusraev ◽  
Z. A. Kusraeva
Keyword(s):  
Linear Operators ◽  
Disjointness Preserving

Orthomorphisms of archimedean vector lattices

1981 ◽  
Vol 89 (1) ◽  
pp. 119-128 ◽  
Author(s):  
S. J. Bernau
Keyword(s):  
Linear Operator ◽  
Vector Lattice ◽  
Jordan Decomposition ◽  
Duality Results ◽  
Vector Lattices ◽  
Elementary Methods ◽  
Disjointness Preserving ◽  
Order Boundedness

AbstractA linear operator T on a vector lattice L preserves disjointness if Tx ⊥ y whenever x ⊥ y. If such a T is positive it is automatically order bounded. An ortho-morphism is an order bounded disjointness preserving linear operator on L. In this note we show that the theory of orthomorphisms on archimedean vector lattices admits a totally elementary exposition. Elementary methods are also effective in duality considerations when the order dual separates points of L. For the Jordan decomposition T = T+ − T− with T+x = (Tx+)+ − (Tx−)+ we can dtrop the order boundedness assumption if we assume either that T preserves ideals or that L is normed and T is continuous. Alternatively we may keep order boundedness and assume only |Tx| ⊥ |Ty| whenever x ⊥ y. The main duality results show: T preserves ideals if and only if T** does; T is an orthomorphism if and only if T* is; T is central (|T| is bounded by a multiple of the identity) if and only if T* is central if and only if T and T* preserve ideals.

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