Schröder Equation in Several Variables and Composition Operator

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.

Power Bounded Composition Operators in Several Variables

Let φ be an analytic self-map of the open unit polydisk D , N ∈ N. Such a map induces a composition operator Cφ acting on weighted Banach spaces of holomorphic functions. We study when such operators are power bounded resp. uniformly mean ergodic. Mathematics Subject Classification (2010): 47B33, 47B38

Composition Operators and Several Complex Variables

Generalized translation operator and approximation in several variables

Generalized translation operators for orthogonal expansions with respect to families of weight functions on the unit ball and on the standard simplex are studied. They are used to define convolution structures and modulus of smoothness for these regions, which are in turn used to characterize the best approximation by polynomials in the weighted Lp spaces. In one variable, this becomes the gene...

Operator monotone functions and Löwner functions of several variables

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...