Maximal spectral type
The type of the maximal spectral measure (i.e. its equivalence class) of a normal operator
acting on a Hilbert space
. This measure is defined (up to equivalence) by the following condition. Let
be the resolution of the identity in the spectral representation of the normal operator
, and let
(where
denotes a Borel set) be the associated "operator-valued" measure. Then
precisely for those
for which
. Any
has an associated spectral measure
; in these terms the definition of
implies that for any
the measure
is absolutely continuous with respect to
and there is an
for which
is equivalent to
(that is,
has maximal spectral type). If
is separable, then a measure
with these properties always exists, but if
is not separable, then there is no such measure and
does not have maximal spectral type. This complicates the theory of unitary invariants of normal operators in the non-separable case.
References
[1] | A.I. Plesner, "Spectral theory of linear operators" , F. Ungar (1969) (Translated from Russian) |
Maximal spectral type. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Maximal_spectral_type&oldid=34116