Self-adjoint Operator - Von Neumann's Formulas

Von Neumann's Formulas

Suppose A is symmetric; any symmetric extension of A is a restriction of A*; Indeed if B is symmetric

Theorem. Suppose A is a densely defined symmetric operator. Let

Then

and

where the decomposition is orthogonal relative to the graph inner product of dom(A*):

These are referred to as von Neumann's formulas in the Akhiezer and Glazman reference.

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