The Spectral And Perturbation Of Two Parameter C Semi-groups

The relationship between operator semi-group and the infinitesimal generator is afundamental problem of operator semi-group theorem. Figuring out the relationshipbetween the two parameters C semi-group and infinitesimal generator is helpful tounderstand two parameters C semi-group and to study the solutions of second orderpartial deferential equations. This paper mainly studies the index formula, spectrum(continuous spectrum, point spectrum, residual spectrum), and operator interference oftwo parameters C semi-group in Banach space. According to the relationship betweentheC0semi-group and two parametersC0semi-group, and the properties of limit andnorm, the definition of C semi-groupis obtained by the study of twoparametersC0semi-group, the index formula, spectrum (continuous spectrum, pointspectrum, residual spectrum), and operator interference are extended to two parametersC semi-group. This paper is consist of the following parts:The first part is the introduction of the concept of two parameters C semi-group andits infinitesimal generator form, the two index formula of two parameters C semi-groupare derived from the properties of two parametersC0semi-group and the form of indexformula.The second part defines the spectrum (continuous spectrum, point spectrum, residualspectrum) of the double parameters C semi-group, and gives the spectrum of twoparameters C semi-group, proves that the spectrum mapping theorem of two parametersC semi-group is authentic.The third part also defines the two parameters C semi-group interference accordingto the two parametersC0semi-group interference and extends it to two parametersC semi-group.