Vol. 56, No. 2, 1975

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On boundary functionals and operators with finite-dimensional null spaces

Franklin Takashi Iha

Vol. 56 (1975), No. 2, 517–524

Let L be a closed operator on a Hilbert space defined on a linear manifold 𝒟 of t with the property that L has a continuous right inverse T and that the dimension of the null space of L is finite. A boundary functional η for L is defined to be a linear functional η on 𝒟 such that ηT is continuous. The boundary-value problems for ordinary differential equations are generalized to the operator L with the boundary conditions defined by a set of boundary functionals. It is shown, in particular, that if K is a continuous right inverse of L, then there exist n linearly independent boundary functionals, η1,n, where n is the dimension of the null space of L, such that the range of K is precisely the linear manifold {u𝜖𝒟|ηI(u) = 0,j = 1,2,,n}.

Mathematical Subject Classification 2000
Primary: 47A05
Secondary: 34B25
Received: 28 November 1973
Published: 1 February 1975
Franklin Takashi Iha