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This paper concerns three classes of real-valued functions on intervals, operator monotone functions, operator convex functions, and strongly operator convex functions. Strongly operator convex functions were previously treated in [3] and [4], where operator algebraic semicontinuity theory or operator theory were substantially used. In this paper we provide an alternate treatment that uses only operator inequalities (or even just matrix inequalities). We show also that if t 0 is a point in the domain of a continuous function f , then f is operator monotone if and only if ( f ( t ) − f ( t 0 ) / ( t − t 0 ) is strongly operator convex. Using this and previously known results, we provide some methods for constructing new functions in one of the three classes from old ones. We also include some discussion of completely monotone functions in this context and some results on the operator convexity or strong operator convexity of φ ∘ f when f is operator convex or strongly operator convex. [ABSTRACT FROM AUTHOR]

The spread of a matrix is defined as the maximum of the distances between any two eigenvalues of that matrix. In this paper we investigate spread maximization as a function on a compact convex subset of the set of real symmetric matrices. We provide some general results and we further study the spread maximizing problem on S n [ a , b ] (the set of symmetric matrices with entries restricted to the interval [ a , b ]). In particular we develop some results by X. Zhan (see [13]), S. M. Fallat and J. J. Xing (see [3]). [ABSTRACT FROM AUTHOR]

Kittaneh proved that if A is a bounded linear operator on a complex Hilbert space, then 1 4 ‖ | A | 2 + | A ⁎ | 2 ‖ ≤ ω 2 (A) , where ω (⋅) and ‖ ⋅ ‖ are the numerical radius and the usual operator norm, and | A | = (A ⁎ A) 1 / 2 . In this paper, we show that 1 4 ‖ | A | 2 + | A ⁎ | 2 ‖ ≤ 1 2 ω 2 (A) + 1 8 ‖ (A + A ⁎) (A − A ⁎) ‖ ≤ ω 2 (A). Meanwhile, we give an improvement of the norm inequality presented by Bhatia and Kittaneh for the positive operators. [ABSTRACT FROM AUTHOR]

In this paper, for a given Eaton triple (V , G , D) , we present a characterization of G -increasing real functions by using the maximality property of their directional derivatives. It generalizes Schur-Ostrowski's condition related to Schur-convexity of a function. It is also useful in establishing theorems of von Neumann-Davis type regarding the form of group-invariant real functions with certain kinds of convexity. We introduce and investigate weak Eaton triples and their chains. We show a method for constructing new Eaton triples. We apply it to define generalized arrangement-increasing (AI) functions. We prove that the mentioned derivatives of G -increasing maps are generalized AI functions. [ABSTRACT FROM AUTHOR]

In this paper, the notion of [ S ] -lineability (originally coined by Vladimir I. Gurariy) is introduced and developed in a general abstract setting. This new notion is, then, applied to specific situations, as for instance, classes of differentiable nowhere monotone functions as well as families of vectors having dense orbit with respect to an operator. Large convex structures are also shown to exist inside the family of topologically mixing continuous selfmaps of a real compact interval. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the question of when the equations A ⁎ s A s = ( A ⁎ A ) s , s ∈ S , where S is a finite set of positive integers, imply the quasinormality or normality of A . In particular, it is proved that if S = { p , m , m + p , n , n + p } , where p ⩾ 1 and 2 ⩽ m < n , then A is quasinormal. Moreover, if A is invertible and S = { m , n , n + m } with m ⩽ n , then A is normal. The case when S = { m , m + n } and A ⁎ n A n ⩽ ( A ⁎ A ) n is also discussed. [ABSTRACT FROM AUTHOR]

In Mousavi and Salemi (2015) [17] , authors showed that joint higher rank numerical range of triple of Pauli group elements is often convex, and characterized it exactly in the case that it is convex. In the present paper, those results are derived in a different manner. Also, N -qubit Pauli group using simultaneously unitary similarity are classified, which may generally be used to characterize joint higher rank numerical range of any Hermitian m -tuple in the N -qubit Pauli group. Besides, some more results about joint numerical range are here derived. [ABSTRACT FROM AUTHOR]

Let C 1 , C 2 , … , C k be positive matrices in M n and f be a continuous real-valued function on [ 0 , ∞ ) . In addition, consider Φ as a positive linear functional on M n and define ϕ ( t 1 , t 2 , t 3 , … , t k ) = Φ ( f ( t 1 C 1 + t 2 C 2 + t 3 C 3 + … + t k C k ) ) , as a k variables continuous function on [ 0 , ∞ ) × … × [ 0 , ∞ ) . In this paper, we show that if f is an operator convex function of order mn , then ϕ is a k variables operator convex function of order ( n 1 , … , n k ) such that m = n 1 n 2 … n k . Also, if f is an operator monotone function of order n k + 1 , then ϕ is a k variables operator monotone function of order n . In particular, if f is a non-negative operator decreasing function on [ 0 , ∞ ) , then the function t → Φ ( f ( A + t B ) ) is an operator decreasing and can be written as a Laplace transform of a positive measure. [ABSTRACT FROM AUTHOR]

In this paper we study some aspects of oblique duality between finite sequences of vectors F and G lying in finite dimensional subspaces W and V , respectively. We compute the possible eigenvalue lists of the frame operators of oblique duals to F lying in V . We compute the spectral and geometrical structure of minimizers of convex potentials among oblique duals for F with norm restrictions; as an application, we show that these optimal duals are the closest to being tight frames and therefore have their spectrum as concentrated as possible, among oblique duals with norm restrictions. We obtain a complete quantitative analysis of the impact that the relative geometry between the subspaces V and W has in oblique duality. We apply this analysis to compute those rigid rotations U for W such that the canonical oblique dual of U ⋅ F minimize every convex potential; we also introduce a notion of aliasing for oblique dual pairs and compute those rigid rotations U for W such that the canonical oblique dual pair associated to U ⋅ F minimize the aliasing. We point out that these two last problems are intrinsic to oblique duality, within the context of consistent sampling. [ABSTRACT FROM AUTHOR]

In this paper, we study a new L 2 norm preserving heat flow in matrix geometry. We show that if the initial data has trace zero and has unit L 2 norm, this flow has a global solution and enjoys the entropy stability in any finite time. We show that as the time is approaching infinity, the flow has its limit as an eigen-matrix of the Laplacian operator. Interesting operator convex property of heat equation is also derived. [ABSTRACT FROM AUTHOR]

In this paper we consider recent work linking “convex majorization” and isotonic operators on ℓ p ( I ) , where I is assumed to be an infinite set and p ∈ [ 1 , + ∞ ) . We proved that P c , the set of all isotonic operators on ℓ p ( I ) is closed. We also provided some other properties about isotonic operators. [ABSTRACT FROM AUTHOR]

Let E be a two-dimensional real normed space. In this paper we show that if the unit circle of E does not contain any line segment such that the distance between its endpoints is greater than 1, then every transformation ϕ : E → E which preserves the unit distance is automatically an affine isometry. In particular, this condition is satisfied when the norm is strictly convex. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a map ( T , B ) → T f T − ( B ) , called an f-connection , induced by an operator convex (concave) function f , where T − denotes a reflexive generalized inverse of a positive linear map T between unital C ⁎ -algebras A and B of Hilbert space operators, and B ∈ B . Some special cases of f -connection with invertible T are related to geometric operator mean, relative operator entropy and Csiszár operator f -divergence. We formulate conditions under which the inequality f ( 1 ) I ≤ ∑ k = 1 n T k f T k − ( B k ) holds, where T k : A → B are positive linear maps and B k ∈ Ran T k . In particular, the following Shannon like inequality 0 ≤ ∑ k = 1 n T k f T k − ( B k ) is shown for an operator convex function f with f ( 1 ) = 0 . The obtained results are specified for Csiszár operator f -divergence. Some recent results by Isa et al. (2013) [15] are recovered. [ABSTRACT FROM AUTHOR]

In this paper, Jensen–Mercer's inequality is generalized by applying the method of pre-majorization used for comparing two tuples of self-adjoint operators. A general result in a matrix setting is established. Special cases of the main theorem are studied to recover other inequalities of Mercer type. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give a method to extract concrete inequalities from a general theorem, which is established by making use of majorization relation between functions. By this method we can get a lot of inequalities; among others we extend Furuta inequality as follows: Let f i , g j be positive operator monotone functions on [ 0 , ∞ ) and put k ( t ) = t r 0 f 1 ( t ) r 1 ⋯ f m ( t ) r m , h ( t ) = t p 0 g 1 ( t ) p 1 ⋯ g n ( t ) p n , where p 0 ≥ 1 and r i ≥ 0 , p j ≥ 0 . Then 0 ≤ A ≤ C ≤ B implies, for 0 < α ≤ 1 + r 0 p + r 0 with p = p 0 + ⋯ + p n , ( k ( C ) 1 2 h ( A ) k ( C ) 1 2 ) α ≤ ( k ( C ) 1 2 h ( C ) k ( C ) 1 2 ) α ≤ ( k ( C ) 1 2 h ( B ) k ( C ) 1 2 ) α . Moreover, we show log ⁡ C 1 / 2 e A C 1 / 2 ≤ log ⁡ C 1 / 2 e C C 1 / 2 ≤ log ⁡ C 1 / 2 e B C 1 / 2 , provided C is invertible. We also refer to operator geometric mean. [ABSTRACT FROM AUTHOR]