# Normal operator

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A closed linear operator defined on a linear subspace that is dense in a Hilbert space such that , where is the operator adjoint to . If is normal, then and for every . Conversely, these conditions guarantee that is normal. If is normal, then so are ; for any ; when it exists; and if , where is a bounded linear operator, then also .

A normal operator has:

1) the multiplicative decomposition  where is a unitary operator which is uniquely determined on the orthogonal complement of the null space of and ; where and are uniquely determined self-adjoint commuting operators.

The additive decomposition implies that for an ordered pair there exists a unique two-dimensional spectral function , where is a two-dimensional interval, , , such that The same decomposition also implies that a normal operator is a function of a certain self-adjoint operator , . Conversely, every function of some self-adjoint operator is normal.

An important property of a normal operator is the fact that , which implies that the spectral radius of a normal operator is its norm . Eigen elements of a normal operator corresponding to distinct eigen values are orthogonal.

How to Cite This Entry:
Normal operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_operator&oldid=17283
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article