Construction Of Complete Lagrangian Subspaces And Dissipative Subspaces In Complex Symplectic Spaces

In this paper,we study the construction of complete Lagrangian subspaces and dissipative subspaces in complex symplectic spaces.There is a one-to-one correspondence between the self-adjoint extensions and dissipative extensions of symmetric operators and the complete Lagrangian subspaces and dissipative subspaces of complex symplectic spaces constructed by the domain of operators.Therefore,the study of the construction of complete Lagrangian subspaces and dissipative subspaces in complex symplectic spaces is of the same significance as the study of the self-adjoint extensions and dissipative extensions of symmetric differential operators.Firstly,we study the construction of complete Lagrangian subspaces in complex symplectic spaces,according to the definition of Lagrangian elements,the necessary and sufficient condition for an element to be Lagrangian element is given.Secondly,the necessary and sufficient condition for two Lagrangian elements to be symplectically orthogonal is given.Finally,the necessary and sufficient condition for a subspace to be a complete Lagrangian subspace is given.In addition,the construction of dissipative subspaces in complex symplectic spaces is researched in this paper.The construction of dissipative subspaces in complex symplectic spaces is considered in general and low order cases,and the sufficient and necessary condition for the subspace of complex symplectic spaces to be dissipative subspace is given.