Symmetric operator

A linear mapping of a set in a Hilbert space (in general, complex) into such that for all . If is an everywhere-dense linear manifold in (and this is assumed in what follows), then is a linear operator. If , then is bounded and hence continuous on . A symmetric operator induces a bilinear Hermitian form on , that is, . The corresponding quadratic form is real. Conversely, if the form on is real, then is symmetric. The sum of two symmetric operators and with a common domain of definition is again a symmetric operator, while if is a real number, then is also symmetric. Every symmetric operator has a uniquely defined closure and an adjoint . In general, is not symmetric and . If , then is called a self-adjoint operator. This holds, for example, in the case of symmetric operators defined on the whole of . If is symmetric and bounded on , then can be extended as a bounded symmetric operator to the whole of .

Examples.

1) Let , be an infinite matrix such that , and

Then the system of equations

defining for an , defines a bounded symmetric operator, which turns out to be self-adjoint on the complex space .

2) In the complex space , let be defined on the set of absolutely-continuous functions on having a square-summable derivative and satisfying the condition . Then is symmetric but not self-adjoint.

References

[1]	L.A. [L.A. Lyusternik] Liusternik, "Elements of functional analysis" , F. Ungar (1961) (Translated from Russian)
[2]	F. Riesz, B. Szökevalfi-Nagy, "Leçons d'analyse fonctionelle" , Akad. Kiado (1955)

Comments

An important problem is to find a self-adjoint extension of a symmetric operator. This problem has different versions, depending on whether one looks for an extension in the original or in a larger space. A complete theory of this topic exists.

References

[a1]	N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1–2 , Pitman (1980) (Translated from Russian)