Fréchet derivatives for 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.
متن کاملApplication of operator monotone functions in economics
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 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...
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In many inverse problems a functional of u is given by measurements where u solves a partial differential equation of the type L(p)u + Su = q. Here, q is a known source term and L(p), S are operators with p as unknown parameter of the inverse problem. For the numerical reconstruction of p often the heuristically derived Fréchet derivative R′ of the mapping R : p → ’measurement functional of u’ ...
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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: 2014
ISSN: 0024-3795
DOI: 10.1016/j.laa.2013.05.018