Maps Preserving Bregman Divergences And Pseudospectra

Let H be Hilbert space over C and B(H)contains all the bounded linear oper-ators on H.This paper focuses on the mapping on B(H)which preserves the skew Jordan product of pseudospectra,and the bijection on positive operators which p-reserves Bregman f divergence(where f is a strictly convex function defined on the positive half axis)in order to summarize the properties of pseudospectra and Breg-man f divergence,and improve the preserver problem on pseudospectra and Breg-man f divergence.The main results are as follows.In the first part,we determine the structure of bijective maps on positive operators preserving the Bregman f-divergence corre-sponding to a differentiable strictly convex function f defined on the positive halfline.It turns out that any such transformation is implemented by either a unitary or an anti-unitary matrix.The second part describes the maps on pseudospectrum pre-serving skew Jordan product.It turns out that for any operator A,there exists a unitary and ? ?{-1,1},such that the certain map is of the form?(A)=?UAU*or ?(A)=?UAtU*,where At denotes the transpose of A,relative to an arbitrary but fixed orthonormal basis of H.