Nonlinear preservers problem of complex symmetric operators

Abstract

Given a conjugation [Formula: see text] on a separable complex Hilbert space [Formula: see text], a bounded linear operator [Formula: see text] on [Formula: see text] is said to be [Formula: see text]-symmetric if [Formula: see text]. In this paper, we study some nonlinear preserving problems concerning [Formula: see text]-symmetric operators and diagonal operators, as a result, we also describe those nonlinear preservers of the lattice of invariant subspaces.

On maps preserving the Jordan product of $C$-symmetric operators

“…Assume the contrary, and let v ∈ u ⊥ be any unit vector. It follows by the previous steps that 1 , and consequently α v,1 = µ v,1 , the desired contradiction. Therefore, A u,λ = 0, and Φ(λu⊗u) = α u,λ u⊗u with α 2 u,λ ̸ = β u,λ = 0 as stated.…”

On maps preserving skew symmetric operators

“…Recently, in [2] the authors considered a non-linear preserver problem involving complex symmetric operators. More precisely, they showed that if Φ is a map on B(H) satisfying…”

“…In this paper, we propose an analogue study for skew symmetric operators. Our arguments are influenced by ideas from [2] and the approaches given therein, but the proofs of our main results require new ingredients. The fundamental results of this paper can be stated as follow: Theorem 1.1.…”