In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, or more specifically in
spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result o ...
, the Riesz projector is the projector onto the eigenspace corresponding to a particular
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of an operator (or, more generally, a projector onto an
invariant subspace In mathematics, an invariant subspace of a linear mapping ''T'' : ''V'' → ''V '' i.e. from some vector space ''V'' to itself, is a subspace ''W'' of ''V'' that is preserved by ''T''; that is, ''T''(''W'') ⊆ ''W''.
General descrip ...
corresponding to an isolated part of the spectrum). It was introduced by
Frigyes Riesz
Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathema ...
in 1912.
[
]
Definition
Let
be a
closed linear operator
In mathematics, particularly in functional analysis and topology, closed graph is a property of functions.
A function between topological spaces has a closed graph if its graph is a closed subset of the product space .
A related property is ...
in the Banach space
. Let
be a simple or composite rectifiable contour, which encloses some region
and lies entirely within the
resolvent set 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
Le ...
(
) of the operator
. Assuming that the contour
has a positive orientation with respect to the region
, the Riesz projector corresponding to
is defined by
:
here
is the
identity operator
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), a ...
in
.
If
is the only point of the spectrum of
in
, then
is denoted by
.
Properties
The operator
is a projector which commutes with
, and hence in the decomposition
:
both terms
and
are
invariant subspaces
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
of the operator
.
Moreover,
# The spectrum of the restriction of
to the subspace
is contained in the region
;
# The spectrum of the restriction of
to the subspace
lies outside the closure of
.
If
and
are two different contours having the properties indicated above, and the regions
and
have no points in common, then the projectors corresponding to them are mutually orthogonal:
:
See also
*
Spectrum (functional analysis)
In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number \lambda is ...
*
Decomposition of spectrum (functional analysis)
The spectrum of a linear operator T that operates on a Banach space X (a fundamental concept of functional analysis) consists of all scalars \lambda such that the operator T-\lambda does not have a bounded inverse on X. The spectrum has a standar ...
*
Spectrum of an operator
In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number \lambda is ...
*
Resolvent formalism In mathematics, the resolvent formalism is a technique for applying concepts from complex analysis to the study of the spectrum of operators on Banach spaces and more general spaces. Formal justification for the manipulations can be found in the fr ...
*
Operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operat ...
References
{{SpectralTheory
Spectral theory