General preservers of invariant subspace lattices

Maps preserving fixed points of generalized product of operators

“…This result has been extended in [3], where G. Dolinar et al characterised the form of maps preserving the lattice of sum of operators, they showed that maps (not necessarily linear) φ : B(X) → B(X) satisfied Lat(φ(A) + φ(B)) = Lat(A + B) for all A, B ∈ B(X), if and only if there is a non zero scalar α and a map ϕ : B(X) → K such that φ(A) = αA + ϕ(A)I for all A ∈ B(X). They proved also, in the same paper, that a non necessarily linear maps φ :…”

“…The problem of characterizing maps on matrices or operators that preserve certain functions, subsets and relations has attracted the attention of many mathematicians in the last decade; for example we can see [1,2,3,4,5,6,7] and their references. In recent years, a great activity has occurred in characterising maps preserving the subspace of fixed points of a matrix or operators.…”

Maps preserving the local spectral subspace of skew-product of operators

“…Their results motivated several authors to describe maps on matrices or operators that preserve local spectrum, local spectral radius, and local inner spectral radius; see, for instance, the last section of the survey article [5] and the references therein. Based on the results from the theory of linear preservers proved by Jafarian and Sourour [14], Dolinar et al [9], characterised the form of maps preserving the lattice of sum of operators. They showed that the map (not necessarily linear) ϕ : B(X) → B(X) satisfies Lat(ϕ(T )+ ϕ(S)) =Lat(T + S) for all T, S ∈ B(X), if and only if there are a non zero scalar α and a map φ : B(X) → F such that ϕ(T ) = αT + φ(T )I for all T ∈ B(X) (See [9, Theorem 1]), where F is the complex field C or the real field R and Lat(T ) is denoted the lattice of T , that is, the set of all invariant subspaces of T .…”

“…Lat(ϕ(T )ϕ(S)ϕ(T )) = Lat(T ST ), resp. Lat(ϕ(T )ϕ(S) + ϕ(S)ϕ(T )) = Lat(T S + ST )) for all T, S ∈ B(X), if and only if there is a map φ : B(X) → F such that ϕ(T ) = 0 if T = 0 and ϕ(T ) = φ(T )T for all T ∈ B(X) (See [9,Theorem 2]).…”

General Preservers of Quasi-Commutativity

“…Among the first examples in this area, we mention Hua's result on adjacency preserving maps [13]. Recently the area of general preservers i.e., not necessarily linear ones, has become very active, see, for example, [1,5,7,15,17,20,26].…”