Rohollah Parvinianzadeh
Jumakhan Pazhman

We collection some results about maps on the algebra of all bounded operators that preserve the local spectrum and local spectral radius at nonzero vectors. Also, we described maps that preserve operators of local spectral radius zero at points and discuss several problems in this direction. Finally, we collection maps that preserve the local spectral subspace of operators associated with any singleton.  

Local spectrum, Local spectral radius, Local spectral subspace, Nonlinear preservers, Matrices.