If f is super - multiplicative , then. The desired inequality 5. Kantorovic, LV: Functional analysis and applied mathematics. Nauk 3 , in Russian. Zhang, F: Equivalence of the Wielandt inequality and the Kantorovich inequality.
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Linear Multilinear Algebra 48 , Springer, Berlin Fiedler, M: Uber eine ungleichung fur positiv definite matrizen. Pure Appl.
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Dragomir, SS: New inequalities of the Kantorovich type for bounded linear operators in Hilbert spaces. Linear Algebra Appl. Springer, New York Theory 73 1 , Moslehian, MS: Recent developments of the operator Kantorovich inequality. Pedersen, GK: Analysis Now. Kubo, F, Ando, T: Means of positive linear operators. Hiai, F: Matrix analysis: matrix monotone functions, matrix means, and majorizations.
Positive Linear Maps of Operator Algebras | Erling Størmer | Springer
Springer, New Delhi Arlinskii, YM: Theory of operator means. Monographs in Inequalities, vol. Element, Zagreb Download references. The author appreciates referees for valuable suggestions which improve the presentation of the paper. Correspondence to Pattrawut Chansangiam. Reprints and Permissions.
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We thus have a very useful technique for studying positive maps. In order to group together maps with similar properties we introduce certain convex cones of positive maps of the bounded operators B H on a Hilbert space into itself, called mapping cones. For example we see that positivity for a map with respect to the smallest mapping cone is equivalent to its dual functional being a positive multiple of a separable state.
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We conclude by showing a Hahn-Banach type extension theorem for maps positive with respect to a mapping cone. A cone in a Hilbert space has a natural dual cone consisting of vectors with positive inner product with vectors in the given cone. In the finite dimensional case we can do the same for cones of positive maps with respect to the Hilbert-Schmidt structure. Characterizations of maps in dual cones are given both in terms of their Choi matrices and their dual functionals.
Furthermore, examples of dual cones of well known mapping cones like for example k -positive maps, are shown to be well known mapping cones. The ideas of dual cones are used in Sect. In this chapter we apply the results on dual functionals of positive maps to study states on tensor products of full matrix algebras. In particular we obtain characterizations of separable and PPT-states, and we also see how entanglement is related to the negative part of the Choi matrix of a positive map.
Finally it is shown that the dual functional of the sum of the trace and a positive map of norm 1 is separable. There are several different norms that can be introduced to positive maps. In this chapter we shall study some, which are closely related to mapping cones, and we show how positivity properties are reflected in norm properties of the maps.