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.
Given a conjugation C on a separable complex Hilbert space H, a bounded
linear operator T on H is said to be C-skew symmetric if CTC = -T*. This
paper describes the maps, on the algebra of all bounded linear operators
acting on H, that preserve the difference of C-skew symmetric operators for
every conjugation C on H.
For a separable complex Hilbert space H, we say that a bounded linear operator T acting on H is C-normal, where C is a conjugation on H, if it satisfies CT * TC = TT * . For a normal operator, we give geometric conditions which guarantee that its rank-one perturbation is a C-normal for some conjugation C.
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