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Real linear maps preserving some complex subspaces. (English) Zbl 1316.15035

Given an invertible real linear map \(A:({V_1},{J_1}) \to ({V_2},{J_2})\), where \({V_1}\) and \({V_2}\) are vector spaces and \({J_1}\) and \({J_2}\) are complex structure operators, the main result of this paper shows that “for a fixed integer \(l\), if \(A:({V_1},{J_1}) \to ({V_2},{J_2})\) has rank greater than 2\(l\) and each of the \({J_1}\)-invariant subspaces with (real) dimension 2\(l\) in some (possibly finite) configuration is mapped into a \({J_2}\)-invariant subspace with dimension 2\(l\), then \(A\) is complex linear \((A \circ {J_1} = {J_2} \circ A)\) or antilinear \((A \circ {J_1} = - {J_2} \circ A)\)”.

MSC:

15A86 Linear preserver problems
15A04 Linear transformations, semilinear transformations
15A24 Matrix equations and identities
14N20 Configurations and arrangements of linear subspaces
53A60 Differential geometry of webs
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