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Jensen's Operator Inequality  [PDF]
Frank Hansen,Gert K. Pedersen
Mathematics , 2002, DOI: 10.1112/S0024609303002200
Abstract: We establish what we consider to be the definitive versions of Jensen's operator inequality and Jensen's trace inequality for functions defined on an interval. This is accomplished by the introduction of genuine non-commutative convex combinations of operators, as opposed to the contractions used in earlier versions of the theory. As a consequence, we no longer need to impose conditions on the interval of definition. We show how this relates to the pinching inequality of Davis, and how Jensen's trace inequlity generalizes to C*-algebras..
Operator inequalities of Jensen type  [PDF]
M. S. Moslehian,J. Micic,M. Kian
Mathematics , 2013,
Abstract: We present some generalized Jensen type operator inequalities involving sequences of self-adjoint operators. Among other things, we prove that if $f:[0,\infty) \to \mathbb{R}$ is a continuous convex function with $f(0)\leq 0$, then {equation*} \sum_{i=1}^{n} f(C_i) \leq f(\sum_{i=1}^{n}C_i)-\delta_f\sum_{i=1}^{n}\widetilde{C}_i\leq f(\sum_{i=1}^{n}C_i) {equation*} for all operators $C_i$ such that $0 \leq C_i\leq M \leq \sum_{i=1}^{n} C_i $ \ $(i=1,...,n)$ for some scalar $M\geq0$, where $ \widetilde{C_i} = 1/2 - |\frac{C_i}{M}- 1/2 |$ and $\delta_f = f(0)+f(M) - 2 f(\frac{M}{2})$.
Jensen's operator inequality and its converses  [PDF]
Frank Hansen,Josip Pecaric,Ivan Peric
Mathematics , 2006,
Abstract: We give a general formulation of Jensen's operator inequality for unital fields of positive linear mappings, and we consider different types of converse inequalities.
Stability of Functional Inequalities with Cauchy-Jensen Additive Mappings  [PDF]
Young-Sun Cho,Hark-Mahn Kim
Abstract and Applied Analysis , 2007, DOI: 10.1155/2007/89180
Abstract: We investigate the generalized Hyers-Ulam stability of the functional inequalities associated with Cauchy-Jensen additive mappings. As a result, we obtain that if a mapping satisfies the functional inequalities with perturbation which satisfies certain conditions, then there exists a Cauchy-Jensen additive mapping near the mapping.
Extension of Jensen's Inequality for Operators without Operator Convexity
Jadranka Mi i ,Zlatko Pavi ,Josip Pe ari
Abstract and Applied Analysis , 2011, DOI: 10.1155/2011/358981
Abstract: We give an extension of Jensen's inequality for -tuples of self-adjoint operators, unital -tuples of positive linear mappings, and real-valued continuous convex functions with conditions on the operators' bounds. We also study operator quasiarithmetic means under the same conditions.
Cauchy-Jensen additive mappings in quasi-Banach algebras and its applications  [PDF]
Abbas Najati,Choonkil Park
Journal of Nonlinear Analysis and Application , 2013, DOI: 10.5899/2013/jnaa-00191
Abstract: In this paper, we prove the Hyers-Ulam stability of homomorphisms in quasi-Banach algebras and of generalized derivations on quasi-Banach algebras for the following Cauchy-Jensen additive mappings
An Operator Hilbert space without the operator approximation property  [PDF]
Alvaro Arias
Mathematics , 2000,
Abstract: We use a technique of Szankowski to construct an operator Hilbert space that does not have the operator approximation property
Jensen and Minkowski inequalities for operator means and anti-norms  [PDF]
Jean-Christophe Bourin,Fumio Hiai
Mathematics , 2011,
Abstract: Jensen inequalities for positive linear maps of Choi and Hansen-Pedersen type are established for a large class of operator/matrix means. These results are also extensions of the Minkowski determinantal inequality. To this end we develop the study of anti-norms, a notion parallel to the symmetric norms in matrix analysis, including functionals like Schatten q-norms for a parameter q<1 and the Minkowski functional. An interpolation theorem for the Schur multiplication is given in this setting. Two sections have been added to the previous version, devoted to means of severable variables and anti-norms.
The Additive Operator Preserving Birkhoff Orthogonal  [PDF]
Sanxing Guo
Journal of Applied Mathematics and Physics (JAMP) , 2019, DOI: 10.4236/jamp.2019.73036
Abstract: The Birkhoff orthogonal plays an important role in the geometric study of Banach spaces. It has been con rmed that a Birkhoff orthogonality preserving linear operator between two normed linear spaces must necessarily be a scalar multiple of a linear isometry. In this paper, the author gives a new result that a Birkhoff orthogonality preserving additive operator between two-dimensional normed linear spaces is necessarily a scalar multiple of a linear isometry.
Additive domain decomposition operator splittings -- convergence analyses in a dissipative framework  [PDF]
Eskil Hansen,Erik Henningsson
Mathematics , 2015,
Abstract: We analyze temporal approximation schemes based on overlapping domain decompositions. As such schemes enable computations on parallel and distributed hardware, they are commonly used when integrating large-scale parabolic systems. Our analysis is conducted by first casting the domain decomposition procedure into a variational framework based on weighted Sobolev spaces. The time integration of a parabolic system can then be interpreted as an operator splitting scheme applied to an abstract evolution equation governed by a maximal dissipative vector field. By utilizing this abstract setting, we derive an optimal temporal error analysis for the two most common choices of domain decomposition based integrators. Namely, alternating direction implicit schemes and additive splitting schemes of first and second order. For the standard first-order additive splitting scheme we also extend the error analysis to semilinear evolution equations, which may only have mild solutions.
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