Hua’s fundamental theorem of the geometry of hermitian matrices characterizes bijective maps on the space of n × n hermitian matrices preserving adjacency in both directions. The problem of possible improvements has been open for a while. We managed to make three natural improvements for the complex hermitian matrices, i.e. we removed the bijectivity assumption, we replaced the assumption of preserving adjacency in both directions by the assumption of preserving adjacency in one direction only, and we also characterized maps acting between the spaces of hermitian matrices of different sizes.
COBISS.SI-ID: 14901337
The problem of characterizing multiplicative maps on matrices over a principal ideal domain was solved by Jodeit and Lam. Pierce showed that their result does not hold true for matrices over an arbitrary integral domain. The motivation to study multiplicative maps on matrices over an arbitrary division ring comes from the Wedderburn-Artin theorem. We managed to describe the general form of endomorphisms of matrix semigroups over an arbitrary not necessarily commutative division ring.
COBISS.SI-ID: 14651737
Bijective linear maps on the algebra of all bounded linear operators on an infinite-dimensional separable Hilbert space, which preserve similarity of operators, are characterized.
COBISS.SI-ID: 15079257
Assume that a map on a space of n x n hermitian matrices, n>2, preserves commutativity in both directions (no linearity or bijectivity is assumed). Then it is a unitary similarity transformation composed with a locally polynomial map possibly composed by the transposition. The same result holds for injective continuous maps preserving commutativity in one direction only. We give counter-examples showing that these two theorems cannot be improved or extended to the infinite-dimensional case.
COBISS.SI-ID: 14722137
We describe the general form of bijective orthogonality preserving maps on n-dimesional real vector space equipped with a pair of generalized indefinite inner products. The relations between the projective space and vector space versions of this result are examined.
COBISS.SI-ID: 15333721