Normal operator

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)

References

[a1]	J.B. Conway, "Subnormal operators" , Pitman (1981)