engleski

Jensen-type functionals for the operators on a Hilbert space

Jensen's functional, derived from the Jensen inequality was proved to possess the properties of superadditivity and monotonicity on the set of all included weight n-tuples. When instead of the real arguments of this functional - the bounded self-adjoint operators on a Hilbert space are considered, the corresponding Jensen-type functional and its variants are proved to possess the analogous properties. These provide us with the general method for refinements and converses of the existing inequalities for certain operator means (arithmetic, geometric and Heinz means), as well as with certain bounds for the eigenvalues of differences of the analyzed operator means, under the assumption on the operators' compactness.

Jensen's inequality; Hilbert space; positive invertible operator; arithmetic operator mean; geometric operator mean; superadditivity; monotonicity; refinement; converse

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