# Normal operator

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A closed linear operator $A$ defined on a linear subspace $D _ {A}$ that is dense in a Hilbert space $H$ such that $A ^ {*} A = AA ^ {*}$, where $A ^ {*}$ is the operator adjoint to $A$. If $A$ is normal, then $D _ {A ^ {*} } = D _ {A}$ and $\| A ^ {*} x \| = \| A x \|$ for every $x$. Conversely, these conditions guarantee that $A$ is normal. If $A$ is normal, then so are $A ^ {*}$; $\alpha A + \beta I$ for any $\alpha , \beta \in \mathbf C$; $A ^ {-} 1$ when it exists; and if $AB = BA$, where $B$ is a bounded linear operator, then also $A ^ {*} B = BA ^ {*}$.

A normal operator has:

1) the multiplicative decomposition

$$A = U \sqrt {A ^ {*} A } = \sqrt {A ^ {*} A } U ,$$

$$A ^ {*} = U ^ {-} 1 \sqrt {A ^ {*} A } = \sqrt {A ^ {*} A } U ^ {-} 1 ,$$

where $U$ is a unitary operator which is uniquely determined on the orthogonal complement of the null space of $A$ and $A ^ {*}$;

2) the additive decomposition

$$A = A _ {1} + iA _ {2} ,\ \ A ^ {*} = A _ {1} - iA _ {2} ,$$

where $A _ {1}$ and $A _ {2}$ are uniquely determined self-adjoint commuting operators.

The additive decomposition implies that for an ordered pair $( A, A ^ {*} )$ there exists a unique two-dimensional spectral function $E ( \Delta _ \zeta )$, where $\Delta _ \zeta$ is a two-dimensional interval, $\Delta _ \zeta = \Delta _ \xi \times \Delta _ \eta$, $\zeta = \xi + i \eta$, such that

$$A = \int\limits _ {\Delta _ \infty } \zeta dE ( \Delta _ \zeta ),\ \ A ^ {*} = \int\limits _ {\Delta _ \infty } \overline \zeta \; dE ( \Delta _ \zeta ).$$

The same decomposition also implies that a normal operator $A$ is a function of a certain self-adjoint operator $C$, $A = F ( C)$. Conversely, every function of some self-adjoint operator is normal.

An important property of a normal operator $A$ is the fact that $\| A ^ {n} \| = \| A \| ^ {n}$, which implies that the spectral radius of a normal operator $A$ is its norm $\| A \|$. Eigen elements of a normal operator corresponding to distinct eigen values are orthogonal.

#### References

 [1] A.I. Plesner, "Spectral theory of linear operators" , F. Ungar (1965) (Translated from Russian) [2] W. Rudin, "Functional analysis" , McGraw-Hill (1973)