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Maps on matrices that preserve the spectral radius distance

Rajendra Bhatia, Peter Šemrl, A. Sourour (1999)

Studia Mathematica

Let ϕ be a surjective map on the space of n×n complex matrices such that r(ϕ(A)-ϕ(B))=r(A-B) for all A,B, where r(X) is the spectral radius of X. We show that ϕ must be a composition of five types of maps: translation, multiplication by a scalar of modulus one, complex conjugation, taking transpose and (simultaneous) similarity. In particular, ϕ is real linear up to a translation.

Maps on upper triangular matrices preserving zero products

Roksana Słowik (2017)

Czechoslovak Mathematical Journal

Consider 𝒯 n ( F ) —the ring of all n × n upper triangular matrices defined over some field F . A map φ is called a zero product preserver on 𝒯 n ( F ) in both directions if for all x , y 𝒯 n ( F ) the condition x y = 0 is satisfied if and only if φ ( x ) φ ( y ) = 0 . In the present paper such maps are investigated. The full description of bijective zero product preservers is given. Namely, on the set of the matrices that are invertible, the map φ may act in any bijective way, whereas for the zero divisors and zero matrix one can write φ as a composition...

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