This paper is an expository devoted to an important class of real-valued functions introduced by Löwner, namely, operator monotone functions. This concept is closely related to operator convex/concave functions. Various characterizations for such functions are given from the viewpoint of differential analysis in terms of matrix of divided differences. From the viewpoint of operator inequalities, various characterizations and the relationship between operator monotonicity and operator convexity are given by Hansen and Pedersen. In the viewpoint of measure theory, operator monotone functions on the nonnegative reals admit meaningful integral representations with respect to Borel measures on the unit interval. Furthermore, Kubo-Ando theory asserts the correspondence between operator monotone functions and operator means.

Let B(H) be the space of all bounded linear operators on a complex separable Hilbert space H. Bohr inequality for Hilbert space operators asserts that for A, B ∈ B(H) and p, q > 1 real numbers such that 1/p + 1/q = 1, |A + B| 2 p| A| 2 + q|B| 2 with equality if and only if B = (p − 1) A. In this paper, a number of generalizations of Bohr inequality for operators in B(H) are established. Moreover, Bohr inequalities are extended to multiple operators and some related inequalities are obtained. The results in this paper generalize results known so far. The idea of transforming problems in operator theory to problems in matrix theory, which are easy to handle, is the key role.

We derive an iterative procedure for solving a generalized Sylvester matrix equation $AXB+CXD = E$ A X B + C X D = E , where $A,B,C,D,E$ A , B , C , D , E are conforming rectangular matrices. Our algorithm is based on gradients and hierarchical identification principle. We convert the matrix iteration process to a first-order linear difference vector equation with matrix coefficient. The Banach contraction principle reveals that the sequence of approximated solutions converges to the exact solution for any initial matrix if and only if the convergence factor belongs to an open interval. The contraction principle also gives the convergence rate and the error analysis, governed by the spectral radius of the associated iteration matrix. We obtain the fastest convergence factor so that the spectral radius of the iteration matrix is minimized. In particular, we obtain iterative algorithms for the matrix equation $AXB=C$ A X B = C , the Sylvester equation, and the Kalman–Yakubovich equation. We give numerical experiments of the proposed algorithm to illustrate its applicability, effectiveness, and efficiency.

We introduce the notion of Khatri-Rao product for operator matrices acting on the direct sum of Hilbert spaces. This notion generalizes the tensor product and Hadamard product of operators and the Khatri-Rao product of matrices. We investigate algebraic properties, positivity, and monotonicity of the Khatri-Rao product. Moreover, there is a unital positive linear map taking Tracy-Singh products to Khatri-Rao products via an isometry.

We propose a new iterative method for solving a generalized Sylvester matrix equation A1XA2+A3XA4=E with given square matrices A1,A2,A3,A4 and an unknown rectangular matrix X. The method aims to construct a sequence of approximated solutions converging to the exact solution, no matter the initial value is. We decompose the coefficient matrices to be the sum of its diagonal part and others. The recursive formula for the iteration is derived from the gradients of quadratic norm-error functions, together with the hierarchical identification principle. We find equivalent conditions on a convergent factor, relied on eigenvalues of the associated iteration matrix, so that the method is applicable as desired. The convergence rate and error estimation of the method are governed by the spectral norm of the related iteration matrix. Furthermore, we illustrate numerical examples of the proposed method to show its capability and efficacy, compared to recent gradient-based iterative methods.