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For an n-tuple of positive invertible operators on a Hilbert space, we present some variants of Ando–Hiai type inequalities for deformed means from an n-variable operator mean by an operator mean, which is related to the information monotonicity of a certain unital positive linear map. As an application, we investigate the monotonicity of the power mean from the deformed mean in terms of the generalized Kantorovich constants under the operator order. Moreover, we improve the norm inequality for the operator power means related to the Log-Euclidean mean in terms of the Specht ratio. [ABSTRACT FROM AUTHOR]

We present some properties of (not necessarily linear) positive maps between C ∗ -algebras. We first extend the notion of Lieb functions to that of Lieb positive maps between C ∗ -algebras. Then we give some basic properties and fundamental inequalities related to such maps. Next, we study n-positive maps ( n ≥ 2). We show that if for a unital 3-positive map Φ : A ⟶ B between unital C ∗ -algebras and some A ∈ A equality Φ (A ∗ A) = Φ (A) ∗ Φ (A) holds, then Φ (X A) = Φ (X) Φ (A) for all X ∈ A . In addition, we prove that for a certain class of unital positive maps Φ : A ⟶ B between unital C ∗ -algebras, the inequality Φ (α A) ≤ α Φ (A) holds for all α ∈ [ 0 , 1 ] and all positive elements A ∈ A if and only if Φ (0) = 0. Furthermore, we show that if for some α in the unit ball of C or in R + with | α | ≠ 0 , 1 , the equality Φ (α I) = α I holds, then Φ is additive on positive elements of A . Moreover, we present a mild condition for a 6-positive map, which ensures its linearity. [ABSTRACT FROM AUTHOR]

We present a necessary and sufficient condition for the norm-parallelism of bounded linear operators on a Hilbert space. We also give a characterization of the Birkhoff–James orthogonality for Hilbert space operators. Moreover, we discuss the connection between norm-parallelism to the identity operator and an equality condition for the Davis–Wielandt radius. Some other related results are also discussed. [ABSTRACT FROM AUTHOR]

We introduce the relation ρ λ -orthogonality in the setting of normed spaces as an extension of some orthogonality relations based on norm derivatives and present some of its essential properties. Among other things, we give a characterization of inner product spaces via the functional ρ λ . Moreover, we consider a class of linear mappings preserving this new kind of orthogonality. In particular, we show that a linear mapping preserving ρ λ -orthogonality has to be a similarity, that is, a scalar multiple of an isometry. [ABSTRACT FROM AUTHOR]

The non-commutative Stein inequality asks whether there exists a constant Cp, q depending only on p, q such that for all (positive) sequences (xn) in. The validity of (Sp, 2) for 1 < p < ∞ and (Sp, 1) for 1 ⩽ p < ∞ are known. In this paper, we verify (i) (Sp, ∞) for 1 < p ⩽ ∞; (ii) (Sp, p) for 1 ⩽ p < ∞; (iii) (Sp, q) for 1 ⩽ q ⩽ 2 and q < p < ∞. We also present some applications. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to establish a noncommutative analogue of the Efron-Stein inequality, which bounds the variance of a general function of some independent random variables. Moreover, we state an operator version including random matrices, which extends a result of D. Paulin et al., [Ann. Probab.44(5) (2016), 3431–3473]. Further, we state a Steele type inequality in the framework of noncommutative probability spaces. [ABSTRACT FROM AUTHOR]

In this paper, we first investigate the matrix equation in a new fashion, where A, B, C, D and G are arbitrary matrices. Then, we establish some necessary and sufficient conditions for the existence of a solution of , where A, B are arbitrary matrices and C is a Hermitian matrix. In the special case when , we determine the general solution of , where A is an arbitrary matrix and C is a Hermitian matrix. [ABSTRACT FROM AUTHOR]

Let A and B be two positive operators with for positive real numbers be an operator mean and be the adjoint mean of If and is a positive unital linear map, then where and is the Kantorovich constant. In addition, for [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a comprehensive study of operator m-convex functions.Let m ∈ [ 0 , 1 ] {m\in[0,1]} , and J = [ 0 , b ] {J=[0,b]} for some b ∈ ℝ {b\in\mathbb{R}} or J = [ 0 , ∞ ) {J=[0,\infty)}.A continuous function φ : J → ℝ {\varphi\colon J\to\mathbb{R}} is called operator m-convex if for any t ∈ [ 0 , 1 ] {t\in[0,1]} and any self-adjoint operators A , B ∈ 𝔹 ⁢ ( ℋ ) {A,B\in\mathbb{B}({\mathscr{H}})} , whose spectra are contained in J, we have φ ⁢ ( t ⁢ A + m ⁢ ( 1 - t ) ⁢ B ) ≤ t ⁢ φ ⁢ ( A ) + m ⁢ ( 1 - t ) ⁢ φ ⁢ ( B ) {\varphi(tA+m(1-t)B)\leq t\varphi(A)+m(1-t)\varphi(B)}.We first generalize the celebrated Jensen inequality for continuous m-convex functions and Hilbert space operators and then use suitable weight functions to give some weighted refinements.Introducing the notion of operator m-convexity, we extend the Choi–Davis–Jensen inequality for operator m-convex functions.We also present an operator version of the Jensen–Mercer inequality for m-convex functions and generalize this inequality for operator m-convex functions involving continuous fields of operators and unital fields of positive linear mappings.Employing the Jensen–Mercer operator inequality for operator m-convex functions, we construct the m-Jensen operator functional and obtain an upper bound for it. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of completely positive matrix of linear maps on Hilbert A -modules over locally C ∗ -algebras and prove an analogue of Stinespring theorem for it. We show that any two minimal Stinespring representations for such matrices are unitarily equivalent. Finally, we prove an analogue of the Radon–Nikodym theorem for this type of completely positive n × n matrices. [ABSTRACT FROM AUTHOR]