\(p\) dissipative operator (Q1302006)

The authors prove that the generalized positive \(p\)-selfadjoint operators in Banach space satisfy the generalized Schwartz inequality and solve the maximal dissipative extension representation of \(p\)-dissipative operators in Banach space. The paper starts with the introduction of generalized Schwartz inequality of generalized positive selfadjoint operators in generalized semi-inner product space. Next, the generalized indefinite inner product space and generalized Krein space are introduced and some of their properties are obtained. Then the natural boundary of a \(p\) dissipative operator in Banach space is constructed and the maximal dissipative extension representation of a sort of \(p\) dissipative operators in Banach space by the natural boundary space is given. Finally, these results are used to study the relevant Schrödinger operator related to soliton wave and quantum mechanics.