Contractive maps on operator ideals and norm inequalities. II (Q344898)

Let \({\mathcal I}\subset {\mathcal B}(H)\) be an operator ideal, endowed with a unitarily invariant norm \(|||\;\;|||\). The authors study contractive maps in \({\mathcal I}\). Let \(f\) be a scalar map, and \(D=L_X+R_Y:{\mathcal I}\to {\mathcal I}\), where \(L_X\) and \(R_Y\) are left and right multiplication operators, respectively, by \(X,Y\in{\mathcal B}(H)\). \(f\) is said to be contractive if \(f(D):{\mathcal I}\to {\mathcal I}\) is a contraction.NEWLINENEWLINENEWLINENEWLINEThe authors use known integral representations and Dirichlet factorization of specific maps, to show that they are contractive (this approach is not new, and the authors provide references to previous works developing this technique).NEWLINENEWLINENEWLINENEWLINEThe fact that a given map is contractive can be stated in terms of an operator inequality, involving the norm \(||| \;\;|||\). The authors show that several known operator inequalities can be obtained in this fashion. Bur also new inequalities, or more general versions of known inequalities, appear. For instance:NEWLINENEWLINENEWLINENEWLINETheorem 2.12. Let \(A,B\) be selfadjoint operators in \({\mathcal B}(H)\), and \(X\in{\mathcal I}\). Then NEWLINE\[NEWLINE |||X|||\leq |||(1+A^2)^{1/2}X(1+B^2)^{1/2}\pm AXB|||. NEWLINE\]NEWLINEFor part I, see [\textit{Y. Kapil} and \textit{M. Singh}, Linear Algebra Appl. 459, 475--492 (2014; Zbl 1309.47059)].