We prove Lieb type convexity and concavity results for trace functionals associated with positive operator monotone (decreasing) functions certain concave functions, together strictly linear maps of matrices. This gives a partial generalization Hiai's recent work on power by allowing decreasing instead maps.Our proof is based variational formula the Legendre transform, strengthened in previous ...

An important and perhaps interesting topic in nonlinear analysis and convex optimization concerns solving inclusions of the form 0 ∈ A(x), where A is a maximal monotone operator on a Hilbert space H. Its importance in convex optimization is evidenced from the fact that many problems that involve convexity can be formulated as finding zeros of maximal monotone operators. For example, convex mini...

In this paper, we study the existence and uniqueness theorem for solving the generalized operator equation of the form F(x) + G(x) + T(x) ∋ 0, where F is a Fréchet differentiable operator, G is a maximal monotone operator and T is a Lipschitzian operator defined on an open convex subset of a Hilbert space. Our results are improvements upon corresponding results of Uko [Generalized equations and...

Inspired by the recent results in [4] and the concept of metric adjusted skew information introduced by Hansen in [6], we here give a further generalization for Schrödinger-type uncertainty relation applying metric adjusted correlation measure introduced in [6]. We firstly give some notations according to those in [4]. Let Mn(C) and Mn,sa(C) be the set of all n × n complex matrices and all n × ...

Cauchy Problem, II. Here we consider the case in which the second operator is symmetric. Assume B : D V is a linear monotone operator with domain D V , where V is a Hilbert space, and that A : V V is a continuous linear symmetric monotone operator. Denote Ž .1 2 the space V with the seminorm A , by V . Then the injection a V V is continuous, and we have V V . V is a Hilbert space, for a a a whi...