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By Per-Olov Lowdin

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This process is equivalent to determining the orthogonal complement of the closed linear manifold formed by the pairs { A t g , - g } . When the closed linear manifold so formed contains no pairs of the type (0,h } , it is the graph of the operator Att which is the least closed extension of the operator A. By examining the graphs it is easy to see that the following propositions are true : (1) If A has an inverse A - ' , then the closure of A implies that of A - ' . (2) If At has an inverse, then (At)-' = (3) If the operator Att exists, its adjoint is At.

We prove the following theorem: The adjoint operator At is uniquely defined if the domain D of the operator A is everywhere dense. ij. It is a linear functional off, and for some g’s it is a bounded linear functional. If g is such an element, then according to the theorem of F. Riesz there is an element g*, such that (Af,g> = (f,9*) for every f E D. (IV. 1) The adjoint operator At is then defined by the equation Ag = g*. 2) Uniqueness of g* follows from the assumption that D is everywhere dense.

4) are called eigenvectors. An eigenvector X’ = {xii, x 2 j , ... ,x i } satisfies the equations n C k= 1 = Ijxi ( i = 1,2, ... 5) which is written in the compact form HX’ = A j X j . The solutions of the Eq. 3) contain a complete orthonormal system of vectors. When these vectors are introduced to define the axes of a coordinate system in the n-dimensioned space, the Hermitean matrix assumes the following diagonal form : 1 ; j. Anumber occurring more than once is said to be a multiple, or degenerate eigenvalue.

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