## 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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### References:

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