Spectral theory of compact operators
From Encyclopedia of Mathematics
Riesz theory of compact operators
Let be a complex Banach space and
a compact operator on
. Then
, the spectrum of
, is countable and has no cluster points except, possibly,
. Every
is an eigenvalue, and a pole of the resolvent function
. Let
be the order of the pole
. For each
,
is closed, and this range is constant for
. The null space
is finite dimensional and constant for
. The spectral projection
(the Riesz projector, see Riesz decomposition theorem) has non-zero finite-dimensional range, equal to
, and its null space is
. Finally,
.
The respective integers and
are called the index and the algebraic multiplicity of the eigenvalue
.
References
[a1] | H.R. Dowson, "Spectral theory of linear operators" , Acad. Press (1978) pp. 45ff. |
[a2] | N. Dunford, J.T. Schwartz, "Linear operators I: General theory" , Interscience (1964) pp. Sect. VII.4 |
How to Cite This Entry:
Spectral theory of compact operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_theory_of_compact_operators&oldid=18215
Spectral theory of compact operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_theory_of_compact_operators&oldid=18215
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article