Unitarily invariant norm inequalities for elementary operators involving G1 operators

In this paper, motivated by perturbation theory of operators, we present some upper bounds for ⦀f(A)Xg(B)+X⦀⦀f(A)Xg(B)+X⦀ in terms of ⦀|AXB|+|X|⦀ and ⦀f(A)Xg(B)−X⦀⦀f(A)Xg(B)−X⦀ in terms of ⦀|AX|+|XB|⦀, where A,BA,B are G1G1 operators, ⦀⋅⦀⦀⋅⦀ is a unitarily invariant norm and f,gf,g are certain analytic functions. Further, we find some new upper bounds for the Schatten 2-norm of f(A)X±Xg(B)f(A)X±Xg(B). Several special cases are discussed as well.