Singular values of compressions, restrictions and dilations Original Research Article

For an operator A and a subspace image , denote by image the restriction of A to image and by image the compression of A to image . A pair (S,T) of hermitian operators is said to be monotone if there exist two nondecreasing functions f, g and a hermitian Z such that S=f(Z) and T=g(Z). Using this notion: (1) We give a simple analytic characterisation of invertible operators X such that image for all subspaces image , where Sing(·) stands for the sequence of singular values. Equivalently, if A is hermitian we characterise invertible operators X satisfying image for all subspaces image , Eig(·) standing for the sequence of eigenvalues. (2) We prove many inequalities involving monotone pairs of positive operators. For instance we show that image . In some other circumstances, the opposite inequality holds. We also show that pairs of hermitians can be dilated into monotone pairs.