Maps preserving operator pairs whose products are projections

Let B ( H ) be the algebra of all bounded linear operators on a complex Hilbert space H with dim H ⩾ 2 . It is proved that a surjective map φ on B ( H ) preserves operator pairs whose products are nonzero projections in both directions if and only if there is a unitary or an anti-unitary operator U...

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Bibliographic Details
Published inLinear algebra and its applications Vol. 433; no. 7; pp. 1348 - 1364
Main Authors Ji, Guoxing, Gao, Yaling
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier Inc 01.12.2010
Elsevier
Subjects
Algebra
Exact sciences and technology
Linear and multilinear algebra, matrix theory
Mathematical analysis
Mathematics
Operator theory
Product of operators
Projection
Sciences and techniques of general use
Triple Jordan product of two operators
Projection
Triple Jordan product of two operators
Map
Product of operators
Primary 47B49
Invariant
Primary: 47B49
Bounded operator
Hilbert space
Jordan algebra
Linear operator
Complex space
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Summary:Let B ( H ) be the algebra of all bounded linear operators on a complex Hilbert space H with dim H ⩾ 2 . It is proved that a surjective map φ on B ( H ) preserves operator pairs whose products are nonzero projections in both directions if and only if there is a unitary or an anti-unitary operator U on H such that φ ( A ) = λ U ∗ AU for all A in B ( H ) for some constants λ with λ 2 = 1 . Related results for surjective maps preserving operator pairs whose triple Jordan products are nonzero projections in both directions are also obtained. These show that the operator pairs whose products or triple Jordan products are nonzero projections are isometric invariants of B ( H ) .
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2010.05.014