On characterization of operator monotone functions
نویسندگان
چکیده
منابع مشابه
Operator monotone functions of several variables
We propose a notion of operator monotonicity for functions of several variables, which extends the well known notion of operator monotonicity for functions of only one variable. The notion is chosen such that a fundamental relationship between operator convexity and operator monotonicity for functions of one variable is extended also to functions of several variables.
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Operator monotone functions play an important role in economics. We show that 2-monotonicity is equivalent to decreasing relative risk premium, a notion recently introduced in microeconomics. We also show that an operator monotone function is risk vulnerable, a notion introduced by Gollier and Pratt.
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We give a characterization of convex functions in terms of difference among values of a function. As an application, we propose an estimation of operator monotone functions: If A > B ≥ 0 and f is operator monotone on (0,∞), then f(A)−f(B) ≥ f(‖B‖+ )−f(‖B‖) > 0, where = ‖(A−B)−1‖−1. Moreover it gives a simple proof to Furuta’s theorem: If logA > logB for A, B > 0 and f is operator monotone on (0...
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We prove generalizations of Löwner’s results on matrix monotone functions to several variables. We give a characterization of when a function of d variables is locally monotone on d-tuples of commuting self-adjoint n-by-n matrices. We prove a generalization to several variables of Nevanlinna’s theorem describing analytic functions that map the upper half-plane to itself and satisfy a growth con...
متن کاملSome Extensions of Loewner’s Theory of Monotone Operator Functions
Several extensions of Loewner’s theory of monotone operator functions are given. These include a theorem on boundary interpolation for matrix-valued functions in the generalized Nevanlinna class. The theory of monotone operator functions is generalized from scalarto matrix-valued functions of an operator argument. A notion of κ-monotonicity is introduced and characterized in terms of classical ...
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ژورنال
عنوان ژورنال: Linear Algebra and its Applications
سال: 2015
ISSN: 0024-3795
DOI: 10.1016/j.laa.2015.09.022