Normal operator
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
;
2) the additive decomposition
![]() |
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.
References
[1] | A.I. Plesner, "Spectral theory of linear operators" , F. Ungar (1965) (Translated from Russian) |
[2] | W. Rudin, "Functional analysis" , McGraw-Hill (1973) |
Comments
References
[a1] | J.B. Conway, "Subnormal operators" , Pitman (1981) |
Normal operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_operator&oldid=17283