Spectral operator
spectral measure
A bounded linear operator mapping a Banach space
into itself and such that for the
-algebra
of Borel subsets
in the plane there is a resolution of the identity
with the following properties: 1) for any
the projector
reduces
, that is,
and the spectrum
lies in
, where
is the restriction of
to the invariant subspace
; 2) the mapping
is a homeomorphism of
into the Boolean algebra
; 3) all projectors
are bounded, that is,
,
; and 4) the resolution of the identity
is countably additive in the strong topology of
, that is, for any
and any sequence
of pairwise disjoint sets,
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The concept of a spectral operator can be generalized to the case of closed unbounded operators. In 1), the additional requirement is then that the inclusion holds, where
is the domain of definition of
, and
for bounded
.
All linear operators on a finite-dimensional space and all self-adjoint and normal operators on a Hilbert space are spectral operators. For example, the operator
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on ,
, is spectral on
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if the kernel is the Fourier transform of a Borel measure
on the plane of total variation
and is such that
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are bounded linear operators on .
Spectral operators have many important properties, such as:
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If is separable, the point and residual spectra of
are at most countable.
References
[1] | N. Dunford, J.T. Schwartz, "Linear operators. Spectral operators" , 3 , Interscience (1971) |
[2] | N. Dunford, "A survey of the theory of spectral operators" Bull. Amer. Math. Soc. , 64 (1958) pp. 217–274 |
Spectral operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_operator&oldid=48761