Theory Of Semigroups And Maximal Tseng Inverse For One-Sided Coupled Operator Matrices

Based on the space decomposition technique,the maximal Tseng inverse and the associated generalized Schur factorization,we investigate the one-sided coupled operator matrix representation of a linear operator in a product space,and the expression of the maximal Tseng inverse,generator property,stability of associated semigroups,essential spectrum,point spectrum and defect spectrum for one-sided coupled operator matrices.Also,we apply the obtained result to concrete mathematical problems.First,using the maximal Tseng inverse,we study the one-sided coupled operator matrix representation of a linear operator in a product space.A necessary and sufficient condition is given for a closed range operator to have an one-sided coupled operator matrix representation.The applications of this result in a delay equation and in a diffusiontransport system with dynamical boundary conditions are further presented.Next,the maximal Tseng inverse A~? of the one-sided coupled operator matrix (?) is studied,a necessary and sufficient condition is given for A~? to have Banachiewicz-Schur form.If A is a bounded operator,then this result is shown to coincide with that in the literature.As an application of the theory of one-sided coupled operator matrices,we can find the minimum norm solution of an elliptic equation with dynamical boundary conditions.Then,the generator property and stability of semigroup for the one-sided coupled operator matrix (?) are considered.Some necessary conditions and sufficient conditions for A to generate an analytic semigroup are considered in the two cases of L ∈ B(X,Y)and L ∈ B(X,Z),and the stability of the analytic semigroup generated by A is obtained by the spectral properties of one-sided coupled operator matrices.Here Z satisfies the(Z)-condition with respect to D.Besides,some necessary and sufficient conditions for A to generate a contraction semigroup are given.As applications,the existence and stability of the solutions to a class of abstract initial boundary value problems are discussed.Finally,in combination with the generalized Schur factorization and the space decomposition technique,we investigate the left(right)essential spectrum,essential spectrum,point spectrum and defect spectrum of the one-sided coupled operator matrix (?).Under more natural conditions,i.e.,if the range of λ-D is closed,thenλ-A has the generalized Schur factorization(?) where Dλ=λ-D and Δ=λ-A-λBD_λ~?L.On this basis,some sufficient conditions are given for λ-A to be left(right)Fredholm,Fredholm,injective and surjective,and these conclusions are further used to determine the left(right)essential spectra and essential spectra of a delay equation,a diffusion-transport system with dynamical boundary conditions and a wave equation with acoustic boundary conditions.