Complete operator
A generalized wave operator, i.e. a partially isometric operator defined by
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where and
are self-adjoint operators on a separable Hilbert space
,
is an ortho-projector into
, and such that
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Here ,
, is the set of all elements
that are spectrally absolutely continuous with respect to
, i.e. for which the spectral measure
of a set
is absolutely continuous with respect to the Lebesgue measure
.
If the operator , or the analogously defined operator
, exists and is complete, the
(the parts of the operators
on
) are unitarily equivalent. If
and
are self-adjoint operators on
and
, where
and
is real, then
and
exist and are complete.
References
[1] | T. Kato, "Perturbation theory for linear operators" , Springer (1966) pp. Chapt. X Sect. 3 |
Comments
An ortho-projector is usually called and orthogonal projector in the West.
An operator is partially isometric if there is a closed linear subspace
of
such that
for
and
for
, the orthogonal complement of
; the set
is called the initial set of
and
the final set of
.
Complete operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_operator&oldid=46420