J. J. Koliha

Two mappings related to semi-inner products and their applications in geometry of normed linear spaces

In this paper we introduce two mappings associated with the lower and upper semi-inner product (·, ·)i and (·, ·)S and with semi-inner products [·, ·] (in the sense of Lumer) which generate the norm of a real normed linear space, and study properties of monotonicity and boundedness of these mappings. We give a refinement of the Schwarz inequality, applications to the Birkhoff orthogonality, to smoothness of normed linear spaces as well as to the characterization of best approximants.

Power bounded and exponentially bounded matrices

The paper gives a new characterization of eigenprojections, which is then used to obtain a spectral decomposition for the power bounded and exponentially bounded matrices. The applications include series and integral representations of the Drazin inverse, and investigation of the asymptotic behaviour of the solutions of singular and singularly perturbed differential equations. An example is given of localized travelling waves for a system of conservation laws.

Closed Semistable Operators and Singular Differential Equations

We study a class of closed linear operators on a Banach space whose nonzero spectrum lies in the open left half plane, and for which 0 is at most a simple pole of the operator resolvent. Our spectral theory based methods enable us to give a simple proof of the characterization of C0-semigroups of bounded linear operators with asynchronous exponential growth, and recover results of Thieme, Webb and van Neerven. The results are applied to the study of the asymptotic behavior of the solutions to a singularly perturbed differential equation in a Banach space.

The mapping Ψ^p_x,y in normed linear spaces and its applications in the theory of inequalities

In this paper we introduce the mapping Ψx,y (t) = (x, x + ty)p ‖x + ty‖−1 , which is derived from the lower and upper semi-inner product (·, ·)i and (·, ·)s , and study its properties of monotonicity, boundedness and convexity. We give applications to height functions and to inequalities in analysis, including a refinement of the Schwarz inequality. Mathematics subject classification (1991): 46B20, 46B99, 46C20, 46C99, 26D15, 26D99.