Spectrum of an operator
The set of complex numbers
for which the operator
does not have an everywhere-defined bounded inverse. Here,
is a linear operator on a complex Banach space
and
is the identity operator on
. If
is not closed on
, then
, and therefore one usually considers spectra of closed operators (the spectrum of the closure of an operator for operators admitting a closure is sometimes called the closure spectrum).
If is either non-injective or non-surjective, then
. In the first case
is called an eigenvalue of
; the set
of eigenvalues is called the point spectrum. In the second case
is called a point of the continuous spectrum
or the residual spectrum
, depending on whether the subspace
is dense in
or not.
There are also other classifications of the points of a spectrum. For example, , where
consists of approximate eigenvalues (
if there are
with
such that
), and
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Note that , and so
. In perturbation theory, use is made of the limit spectrum
, which consists of the limit points of
and the isolated eigenvalues of infinite multiplicity, of the Weyl spectrum, which is equal to the intersection of the spectra of all compact perturbations, etc.
If is a bounded operator, then
is compact and non-empty (in this case
coincides with the spectrum of the element
of the Banach algebra
, cf. Spectrum of an element); in general one can only say that
is closed in
. On the set
one can define the analytic
-valued function
, called the resolvent of
(
is called the resolvent set). With the help of resolvents a functional calculus for
is built on functions analytic in a neighbourhood of
:
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where is a contour enclosing
(the unboundedness of
imposes restrictions on the choice of
). Further conditions on the geometry of the spectrum and on the asymptotics of the resolvent enables one to extend this calculus.
The spectra of operator functions are defined by the formula
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(the spectral mapping theorem). The spectrum of the adjoint operator coincides with
when
is bounded; in general,
.
If , then
, and
decomposes into the direct sum of subspaces invariant under
, on each of which
induces an operator with one-point spectrum. Spectral theory of operators is concerned with finding infinite-dimensional analogues for this decomposition. See also Spectral analysis; Spectral synthesis; Spectral operator; Spectral decomposition of a linear operator.
References
[1] | N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) |
[2] | T. Kato, "Perturbation theory for linear operators" , Springer (1980) |
Comments
References
[a1] | A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980) |
[a2] | F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French) |
Spectrum of an operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectrum_of_an_operator&oldid=11393