In
mathematics, particularly in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
, the spectrum of a
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
(or, more generally, an
unbounded linear operator In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases.
The ter ...
) is a generalisation of the set of
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 ...
s of a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
. Specifically, a
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
is said to be in the spectrum of a bounded linear operator
if
is not
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
, where
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) ...
. The study of spectra and related properties is known as
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 ...
, which has numerous applications, most notably the
mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which ...
.
The spectrum of an operator on a
finite-dimensional vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
is precisely the set of eigenvalues. However an operator on an infinite-dimensional space may have additional elements in its spectrum, and may have no eigenvalues. For example, consider the
right shift operator ''R'' on the
Hilbert space ℓ2,
:
This has no eigenvalues, since if ''Rx''=''λx'' then by expanding this expression we see that ''x''
1=0, ''x''
2=0, etc. On the other hand, 0 is in the spectrum because the operator ''R'' − 0 (i.e. ''R'' itself) is not invertible: it is not surjective since any vector with non-zero first component is not in its range. In fact ''every'' bounded linear operator on a
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
Banach space must have a non-empty spectrum.
The notion of spectrum extends to
unbounded (i.e. not necessarily bounded) operators. A
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
''λ'' is said to be in the spectrum of an unbounded operator
defined on domain
if there is no bounded inverse
defined on the whole of
If ''T'' is
closed (which includes the case when ''T'' is bounded), boundedness of
follows automatically from its existence.
The space of bounded linear operators ''B''(''X'') on a Banach space ''X'' is an example of a
unital Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
. Since the definition of the spectrum does not mention any properties of ''B''(''X'') except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.
Spectrum of a bounded operator
Definition
Let
be a
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
acting on a Banach space
over the complex scalar field
, and
be 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) ...
on
. The spectrum of
is the set of all
for which the operator
does not have an inverse that is a bounded linear operator.
Since
is a linear operator, the inverse is linear if it exists; and, by the
bounded inverse theorem In mathematics, the bounded inverse theorem (or inverse mapping theorem) is a result in the theory of bounded linear operators on Banach spaces.
It states that a bijective bounded linear operator ''T'' from one Banach space to another has bounded ...
, it is bounded. Therefore, the spectrum consists precisely of those scalars
for which
is not
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
.
The spectrum of a given operator
is often denoted
, and its complement, 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
...
, is denoted
. (
is sometimes used to denote the spectral radius of
)
Relation to eigenvalues
If
is an eigenvalue of
, then the operator
is not one-to-one, and therefore its inverse
is not defined. However, the converse statement is not true: the operator
may not have an inverse, even if
is not an eigenvalue. Thus the spectrum of an operator always contains all its eigenvalues, but is not limited to them.
For example, consider the Hilbert space
, that consists of all
bi-infinite sequences of real numbers
:
that have a finite sum of squares
. The
bilateral shift operator
simply displaces every element of the sequence by one position; namely if
then
for every integer
. The eigenvalue equation
has no nonzero solution in this space, since it implies that all the values
have the same absolute value (if
) or are a geometric progression (if
); either way, the sum of their squares would not be finite. However, the operator
is not invertible if
. For example, the sequence
such that
is in
; but there is no sequence
in
such that
(that is,
for all
).
Basic properties
The spectrum of a bounded operator ''T'' is always a
closed,
bounded and
non-empty
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other t ...
subset of the
complex plane.
If the spectrum were empty, then the
''resolvent function''
:
would be defined everywhere on the complex plane and bounded. But it can be shown that the resolvent function ''R'' is
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
on its domain. By the vector-valued version of
Liouville's theorem, this function is constant, thus everywhere zero as it is zero at infinity. This would be a contradiction.
The boundedness of the spectrum follows from the
Neumann series expansion in ''λ''; the spectrum ''σ''(''T'') is bounded by , , ''T'', , . A similar result shows the closedness of the spectrum.
The bound , , ''T'', , on the spectrum can be refined somewhat. The ''
spectral radius
In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectru ...
'', ''r''(''T''), of ''T'' is the radius of the smallest circle in the complex plane which is centered at the origin and contains the spectrum ''σ''(''T'') inside of it, i.e.
:
The spectral radius formula says
[Theorem 3.3.3 of Kadison & Ringrose, 1983, ''Fundamentals of the Theory of Operator Algebras, Vol. I: Elementary Theory'', New York: Academic Press, Inc.] that for any element
of a
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
,
:
Spectrum of an unbounded operator
One can extend the definition of spectrum to
unbounded operator In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases.
The ter ...
s on a
Banach space ''X''. These operators which are no longer elements in the Banach algebra ''B''(''X'').
Definition
Let ''X'' be a Banach space and
be a
linear operator defined on domain
.
A complex number ''λ'' is said to be in the resolvent set (also called regular set) of
if the operator
:
has a bounded everywhere-defined inverse, i.e. if there exists a bounded operator
:
such that
:
A complex number ''λ'' is then in the spectrum if ''λ'' is not in the resolvent set.
For ''λ'' to be in the resolvent (i.e. not in the spectrum), just like in the bounded case,
must be bijective, since it must have a two-sided inverse. As before, if an inverse exists, then its linearity is immediate, but in general it may not be bounded, so this condition must be checked separately.
By the
closed graph theorem
In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs.
Each gives conditions when functions with closed graphs are necessarily continuous.
Graphs and m ...
, boundedness of
''does'' follow directly from its existence when ''T'' is
closed. Then, just as in the bounded case, a complex number ''λ'' lies in the spectrum of a closed operator ''T'' if and only if
is not bijective. Note that the class of closed operators includes all bounded operators.
Basic properties
The spectrum of an unbounded operator is in general a closed, possibly empty, subset of the complex plane.
If the operator ''T'' is not
closed, then
.
Classification of points in the spectrum
A bounded operator ''T'' on a Banach space is invertible, i.e. has a bounded inverse, if and only if ''T'' is bounded below, i.e.
for some
and has dense range. Accordingly, the spectrum of ''T'' can be divided into the following parts:
#
if
is not bounded below. In particular, this is the case if
is not injective, that is, ''λ'' is an eigenvalue. The set of eigenvalues is called the point spectrum of ''T'' and denoted by ''σ''
p(''T''). Alternatively,
could be one-to-one but still not bounded below. Such ''λ'' is not an eigenvalue but still an ''approximate eigenvalue'' of ''T'' (eigenvalues themselves are also approximate eigenvalues). The set of approximate eigenvalues (which includes the point spectrum) is called the approximate point spectrum of ''T'', denoted by ''σ''
ap(''T'').
#
if
does not have dense range. The set of such ''λ'' is called the compression spectrum of ''T'', denoted by
. If
does not have dense range but is injective, ''λ'' is said to be in the residual spectrum of ''T'', denoted by
.
Note that the approximate point spectrum and residual spectrum are not necessarily disjoint (however, the point spectrum and the residual spectrum are).
The following subsections provide more details on the three parts of ''σ''(''T'') sketched above.
Point spectrum
If an operator is not injective (so there is some nonzero ''x'' with ''T''(''x'') = 0), then it is clearly not invertible. So if ''λ'' is an
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 ...
of ''T'', one necessarily has ''λ'' ∈ ''σ''(''T''). The set of eigenvalues of ''T'' is also called the point spectrum of ''T'', denoted by ''σ''
p(''T'').
Approximate point spectrum
More generally, by the
bounded inverse theorem In mathematics, the bounded inverse theorem (or inverse mapping theorem) is a result in the theory of bounded linear operators on Banach spaces.
It states that a bijective bounded linear operator ''T'' from one Banach space to another has bounded ...
, ''T'' is not invertible if it is not bounded below; that is, if there is no ''c'' > 0 such that , , ''Tx'', , ≥ ''c'', , ''x'', , for all . So the spectrum includes the set of approximate eigenvalues, which are those ''λ'' such that is not bounded below; equivalently, it is the set of ''λ'' for which there is a sequence of unit vectors ''x''
1, ''x''
2, ... for which
:
.
The set of approximate eigenvalues is known as the approximate point spectrum, denoted by
.
It is easy to see that the eigenvalues lie in the approximate point spectrum.
For example, consider the right shift ''R'' on
defined by
:
where
is the standard orthonormal basis in
. Direct calculation shows ''R'' has no eigenvalues, but every ''λ'' with , ''λ'', = 1 is an approximate eigenvalue; letting ''x''
''n'' be the vector
:
one can see that , , ''x''
''n'', , = 1 for all ''n'', but
:
Since ''R'' is a unitary operator, its spectrum lies on the unit circle. Therefore, the approximate point spectrum of ''R'' is its entire spectrum.
This conclusion is also true for a more general class of operators.
A unitary operator is
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
. By the
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful ...
, a bounded operator on a Hilbert space H is normal if and only if it is equivalent (after identification of ''H'' with an
space) to a
multiplication operator
In operator theory, a multiplication operator is an operator defined on some vector space of functions and whose value at a function is given by multiplication by a fixed function . That is,
T_f\varphi(x) = f(x) \varphi (x) \quad
for all in th ...
. It can be shown that the approximate point spectrum of a bounded multiplication operator equals its spectrum.
Continuous spectrum
The set of all ''λ'' for which
is injective and has dense range, but is not surjective, is called the continuous spectrum of ''T'', denoted by
. The continuous spectrum therefore consists of those approximate eigenvalues which are not eigenvalues and do not lie in the residual spectrum. That is,
:
.
For example,
,
,
, is injective and has a dense range, yet
.
Indeed, if
with
such that
, one does not necessarily have
, and then
.
Compression spectrum
The set of
for which
does not have dense range is known as the compression spectrum of ''T'' and is denoted by
.
Residual spectrum
The set of
for which
is injective but does not have dense range is known as the residual spectrum of ''T'' and is denoted by
:
:
An operator may be injective, even bounded below, but still not invertible. The right shift on
,
,
, is such an example. This shift operator is an
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
, therefore bounded below by 1. But it is not invertible as it is not surjective (
), and moreover
is not dense in
(
).
Peripheral spectrum
The peripheral spectrum of an operator is defined as the set of points in its spectrum which have modulus equal to its spectral radius.
Discrete spectrum
The
discrete spectrum
A observable, physical quantity is said to have a discrete spectrum if it takes only distinct values, with gaps between one value and the next.
The classical example of discrete spectrum (for which the term was first used) is the characterist ...
is defined as the set of
normal eigenvalue
In mathematics, specifically in spectral theory, an eigenvalue of a closed linear operator is called normal if the space admits a decomposition into a direct sum of a finite-dimensional generalized eigenspace and an invariant subspace where A-\lam ...
s. Equivalently, it can be characterized as the set of isolated points of the spectrum such that the corresponding
Riesz projector In mathematics, or more specifically in spectral theory, the Riesz projector is the projector onto the eigenspace corresponding to a particular eigenvalue of an operator (or, more generally, a projector onto an invariant subspace corresponding to an ...
is of finite rank.
Essential spectrum
There are five similar definitions of the
essential spectrum In mathematics, the essential spectrum of a bounded operator (or, more generally, of a densely defined closed linear operator) is a certain subset of its spectrum, defined by a condition of the type that says, roughly speaking, "fails badly to be ...
of closed densely defined linear operator
which satisfy
:
All these spectra
, coincide in the case of self-adjoint operators.
# The essential spectrum
is defined as the set of points
of the spectrum such that
is not
semi-Fredholm. (The operator is ''semi-Fredholm'' if its range is closed and either its kernel or cokernel (or both) is finite-dimensional.)
Example 1:
for the operator
,
(because the range of this operator is not closed: the range does not include all of
although its closure does).
Example 2:
for
,
for any
(because both kernel and cokernel of this operator are infinite-dimensional).
# The essential spectrum
is defined as the set of points
of the spectrum such that the operator either
has infinite-dimensional kernel or has a range which is not closed. It can also be characterized in terms of ''Weyl's criterion'': there exists a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
in the space ''X'' such that
,
and such that
contains no convergent
subsequence
In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a ...
. Such a sequence is called a ''singular sequence'' (or a ''singular Weyl sequence'').
Example:
for the operator
,
if ''j'' is even and
when ''j'' is odd (kernel is infinite-dimensional; cokernel is zero-dimensional). Note that
.
# The essential spectrum
is defined as the set of points
of the spectrum such that
is not
Fredholm. (The operator is ''Fredholm'' if its range is closed and both its kernel and cokernel are finite-dimensional.)
Example:
for the operator
,
(kernel is zero-dimensional, cokernel is infinite-dimensional). Note that
.
# The essential spectrum
is defined as the set of points
of the spectrum such that
is not
Fredholm of index zero. It could also be characterized as the largest part of the spectrum of ''A'' which is preserved by
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
perturbations. In other words,
; here
denotes the set of all compact operators on ''X''.
Example:
where
is the right shift operator,
,
for
(its kernel is zero, its cokernel is one-dimensional). Note that
.
# The essential spectrum
is the union of
with all components of
that do not intersect with the resolvent set
. It can also be characterized as
.
Example: consider the operator
,
for
,
. Since
, one has
. For any
with
, the range of
is dense but not closed, hence the boundary of the unit disc is in the first type of the essential spectrum:
. For any
with
,
has a closed range, one-dimensional kernel, and one-dimensional cokernel, so
although
for
; thus,
for
. There are two components of
:
and
. The component
has no intersection with the resolvent set; by definition,
.
Example: Hydrogen atom
The
hydrogen atom provides an example of different types of the spectra. The
hydrogen atom Hamiltonian operator ,
, with domain
has a discrete set of eigenvalues (the discrete spectrum
, which in this case coincides with the point spectrum
since there are no eigenvalues embedded into the continuous spectrum) that can be computed by the
Rydberg formula
In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line in many chemical elements. The formula was primarily presented as a generalization of the Balmer series for all atomic electron transitions of hydrogen. It wa ...
. Their corresponding
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s are called eigenstates, or the
bound state
Bound or bounds may refer to:
Mathematics
* Bound variable
* Upper and lower bounds, observed limits of mathematical functions
Physics
* Bound state, a particle that has a tendency to remain localized in one or more regions of space
Geography
* ...
s. The result of the
ionization
Ionization, or Ionisation is the process by which an atom or a molecule acquires a negative or positive charge by gaining or losing electrons, often in conjunction with other chemical changes. The resulting electrically charged atom or molecul ...
process is described by the continuous part of the spectrum (the energy of the collision/ionization is not "quantized"), represented by