MATHEMATICA BOHEMICA, Vol. 121, No. 3, pp. 301-314, 1996

On solvability of nonlinear operator equations and eigenvalues of homogeneous operators

Vera Buryskova, Slavomir Burysek

Vera Buryskova, Slavomir Burysek, Department of Mathematics, Faculty of Mechanical Engineering, Czech Technical University Prague, Karlovo nam. 13, 121 35 Praha 2, Czech Republic

Abstract: Notions as the numerical range $W(S,T)$ and the spectrum $\s(S,T)$ of couple $(S,T)$ of homogeneous operators on a Banach space are used to derive theorems on solvability of the equation $Sx-lTx=y.$ Conditions for the existence of eigenvalues of the couple $(S,T)$ are given.

Keywords: Banach and Hilbert space, homogeneous, polynomial, symmetric, monotone operator, numerical range, spectrum, eigenvalue.

Classification (MSC 1991): 47H15

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