| Spectral theory of operators is one of the most active research topics in operator theory,which has a wide range of applications in many aspects of mathematical physics.In the past 60 years,mathematicians have introduced many definitions of the spectrum for nonlinear operators,which have important applications in nonlinear integral equations,bifurcation theory,boundary value problems and solving nonlinear equations.Recently,the nonlinear block operator matrices begin to attract authors’ attention.This dissertation mainly deals with some properties of the Kachurovskij spectrum,the Furi-Martelli-Vignoli spectrum,the Feng spectrum and the Infante-Webb spectrum of nonlinear block operator matrices.As an application,the numerical range and numerical radius of nonlinear block operator matrices are further studied.The details are as follows:Firstly,the Kachurovskij spectrum of diagonal nonlinear block operator matrices,upper triangular nonlinear block operator matrices and more general 2 × 2 nonlinear block operator matrices are studied respectively,and the relationship between the Kachurovskij spectrum of those nonlinear block operator matrices and that of their entries and that of their Schur complement is obtained.Moreover,the spectral inclusion properties of2 × 2 Lipschitz continuous nonlinear block operator matrices are characterized by using the Gershgorin theorem and numerical range,and the lipeomorphism of 2 × 2 Lipschitz continuous nonlinear block operator matrices is studied by using the perturbation theory of nonlinear operators.Secondly,the Furi-Martelli-Vignoli spectrum,the Feng spectrum and the InfanteWebb spectrum of nonlinear block operator matrices are studied.Some connections between the Furi-Martelli-Vignoli spectrum(Feng spectrum and Infante-Webb spectrum)of nonlinear block operator matrices and that of their entries are explored,and the relationship between the Furi-Martelli-Vignoli spectrum(Infante-Webb spectrum)of nonlinear block operator matrices and that of their Schur complement is obtained by means of Forbenius-Schur decomposition.Then,as an application,the numerical range and numerical radius of nonlinear operators and nonlinear block operator matrices are studied.Some conclusions of the numerical range and numerical radius for bounded linear operators are extended to the case of the numerical range and numerical radius for nonlinear operators respectively,and the relationship between the numerical range of nonlinear block operator matrices and that of their diagonal entries is given,furthermore,several numerical radius inequalities of nonlinear block operator matrices are given.Finally,as an important generalization of the numerical radius of bounded linear operators,the Davis-Wielandt radius inequalities of bounded linear operators and offdiagonal block operator matrices are studied.Using the generalized mixed Schwartz inequality,Young inequality,Jensen inequality and some elementary inequalities in inner product space,several upper bounds for the Davis-Wielandt radius of off-diagonal block operator matrices,as well as some upper and lower bounds for the Davis-Wielandt radius of bounded linear operators are obtained. |
