Nonhomogeneous boundary value problems for linear manifolds
Lett \(T_ 0\) and \(T_ 1\) be two closed linear manifolds contained in the direct sum \(X\oplus Y\) of Banach spaces X,Y such that \(T_ 0\subset T_ 1\) and \(N:=\dim T_ 1/T_ 0\) is finite. In a concrete case they might be the graphs of differential operators. Take a closed linear manifold T between the two manifolds and write it as \(T=\{x\in T_ 1| \quad PB(x)=O_{m\times 1}\},\) where B is a continuous linear operator from \(T_ 1\) into the N-dimensional complex Euclidean space and P in an \(m\times N\) matrix of rank m. The main parts of the paper consist of the following: (1) Assume that \(n:=Null T_ 1\) and \(n^*:=\dim Null T^*_ 0\) are finite. If \(N=n+n^*,\) then \[ \dim Null T=\dim Null T^*+N-m-n^*. \] (2) Let g and r be elements in Range \(T_ 1\) and \({\mathbb{C}}^ m\), respectively. We give a necessary and sufficient condition for the existence of an element s in \(Dom T_ 1\) such that \(\{s,g\}\in T_ 1,\quad PB(\{s,g\})^*=r^ t.\) This condition is expressed in terms of \(Null T^*\) and P, \((PP^*)^{-1}\) etc. This theory has been applied to differential operators by the author in Kyungpook Math. J. 23, No.2, 115-128 (1983). The above abstract theory has been generalized by the author to the case when N is infinite [Infinitely many nonhomogeneous boundary conditions, J. Math. Anal. Appl. (in press).]
scientific article; zbMATH DE number 3851836