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Hermitian operator

A linear operator $A$ defined on a linear everywhere-dense set $D ( A)$ in a Hilbert space $H$ and coinciding with its adjoint operator $A ^ {*}$, that is, such that $D ( A) = D ( A ^ {*} )$ and

$$\tag{* } \langle Ax, y \rangle = \langle x, Ay \rangle$$

for every $x, y \in D ( A)$. Every self-adjoint operator is closed and cannot be extended with the preservation of (*) to a linear manifold wider than $D ( A)$; in view of this a self-adjoint operator is also called hypermaximal. Therefore, if $A$ is a bounded self-adjoint operator, then it is defined on the whole of $H$.

Every self-adjoint operator uniquely determines a resolution of the identity $E _ \lambda$, $- \infty < \lambda < \infty$; the following integral representation holds:

$$Ax = \int\limits _ {- \infty } ^ \infty \lambda dE _ \lambda x,$$

where the integral is understood as the strong limit of the integral sums for each $x \in D ( A)$, and

$$D ( A) = \ \left \{ {x } : { \int\limits _ {- \infty } ^ \infty \lambda ^ {2} d \langle E _ \lambda x, x \rangle < \infty } \right \} .$$

The spectrum of a self-adjoint operator is non-empty and lies on the real line. The quadratic form $K ( A) = \langle Ax, x \rangle$ generated by a self-adjoint operator $A$ is real, and this enables one to introduce the concept of a positive operator.

Many boundary value problems of mathematical physics are described by means of self-adjoint operators.

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