Resolvent set explained
In linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved". The resolvent set plays an important role in the resolvent formalism.
Definitions
Let X be a Banach space and let
be a linear operator with
domain
. Let id denote the
identity operator on
X. For any
, let
A complex number
is said to be a
regular value if the following three statements are true:
is
injective, that is, the corestriction of
to its image has an
inverse
called the
resolvent;
is a
bounded linear operator;
is defined on a
dense subspace of
X, that is,
has dense range.The
resolvent set of
L is the set of all regular values of
L:
\rho(L)=\{λ\inC\midλisaregularvalueofL\}.
The spectrum is the complement of the resolvent set
\sigma(L)=C\setminus\rho(L),
and subject to a mutually singular spectral decomposition into the point spectrum (when condition 1 fails), the continuous spectrum (when condition 2 fails) and the residual spectrum (when condition 3 fails).
If
is a
closed operator, then so is each
, and condition 3 may be replaced by requiring that
be
surjective.
Properties
of a bounded linear operator
L is an
open set.
- More generally, the resolvent set of a densely defined closed unbounded operator is an open set.
References
- Book: Reed . M. . Simon . B. . Methods of Modern Mathematical Physics: Vol 1: Functional analysis . Academic Press . 1980 . 978-0-12-585050-6.
- Book: Renardy
, Michael
. Rogers, Robert C. . An introduction to partial differential equations. Texts in Applied Mathematics 13. Second. Springer-Verlag. New York. 2004. 0-387-00444-0. xiv+434. true. (See section 8.3)
See also