关于可闭的K-正定算子方程的注记

Note on Closed of K-positive Definite Operator Equations

设X为任意Banach空间,X*为其共轭空间,A:D(A)X→X*为可闭的K-正定算子,D(A)=D(K),则存在常数α>0使得x∈D(A),有‖Ax‖≤α‖Kx‖,而且A为闭算子,R(A)=X*,f∈X*,方程Ax=f有唯一解.

Let X be an arbitrary Banach space with a dual X^＊ and let A ： D （A） belong to X→ X^＊ be a K-positive definite operator with D （A） = D （K）. Then there exists a constant α 〉 0 such that ‖Ax ‖ ≤ α ‖Kx‖ ,for all x∈D（A）. Furthermore, the operator A is closed, R （A） = X^＊ and the equation Ax = f, for each f∈X^＊ , has a unique solution . In the case X is a Hilbert space, a constructive solvability for the equation Ax = f, for each f∈H, is also given.