Spectrum Of An Operator
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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 ...
, 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 space#Definition, inner product, Norm (mathematics)#Defini ...
, 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 b ...
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 form ...
\lambda is said to be in the spectrum of a bounded linear operator T if T-\lambda I 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 I 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 ...
. 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 o ...
, 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 In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...
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 In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
2, :(x_1, x_2, \dots) \mapsto (0, x_1, x_2, \dots). 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 In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
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 form ...
''λ'' is said to be in the spectrum of an unbounded operator T:\,X\to X defined on domain D(T)\subseteq X if there is no bounded inverse (T-\lambda I)^:\,X\to D(T) defined on the whole of X. If ''T'' is
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
(which includes the case when ''T'' is bounded), boundedness of (T-\lambda I)^ 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 spa ...
. 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 T 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 X over the complex scalar field \mathbb, and I 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), a ...
on X. The spectrum of T is the set of all \lambda \in \mathbb for which the operator T-\lambda I does not have an inverse that is a bounded linear operator. Since T-\lambda I 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 \lambda for which T-\lambda I 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 s ...
. The spectrum of a given operator T is often denoted \sigma(T), 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 Le ...
, is denoted \rho(T) = \mathbb \setminus \sigma(T). (\rho(T) is sometimes used to denote the spectral radius of T)


Relation to eigenvalues

If \lambda is an eigenvalue of T, then the operator T-\lambda I is not one-to-one, and therefore its inverse (T-\lambda I)^ is not defined. However, the converse statement is not true: the operator T - \lambda I may not have an inverse, even if \lambda 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 \ell^2(\Z), that consists of all bi-infinite sequences of real numbers :v = (\ldots, v_,v_,v_0,v_1,v_2,\ldots) that have a finite sum of squares \sum_^ v_i^2. The bilateral shift operator T simply displaces every element of the sequence by one position; namely if u = T(v) then u_i = v_ for every integer i. The eigenvalue equation T(v) = \lambda v has no nonzero solution in this space, since it implies that all the values v_i have the same absolute value (if \vert \lambda \vert = 1) or are a geometric progression (if \vert \lambda \vert \neq 1); either way, the sum of their squares would not be finite. However, the operator T-\lambda I is not invertible if , \lambda, = 1. For example, the sequence u such that u_i = 1/(, i, +1) is in \ell^2(\Z); but there is no sequence v in \ell^2(\Z) such that (T-I)v = u (that is, v_ = u_i + v_i for all i).


Basic properties

The spectrum of a bounded operator ''T'' is always a
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
, 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 In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
. If the spectrum were empty, then the ''resolvent function'' :R(\lambda) = (T-\lambda I)^, \qquad \lambda\in\Complex, 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 derivativ ...
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. :r(T) = \sup \. The spectral radius formula saysTheorem 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 T 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 spa ...
, :r(T) = \lim_ \left\, T^n\right\, ^.


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 In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
''X''. These operators which are no longer elements in the Banach algebra ''B''(''X'').


Definition

Let ''X'' be a Banach space and T:\,D(T)\to X be a
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
defined on domain D(T) \subseteq X. A complex number ''λ'' is said to be in the resolvent set (also called regular set) of T if the operator :T-\lambda I:\,D(T) \to X has a bounded everywhere-defined inverse, i.e. if there exists a bounded operator :S :\, X \rightarrow D(T) such that :S (T - \lambda I) = I_, \, (T - \lambda I) S = I_X. 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, T-\lambda I 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 (T-\lambda I)^ ''does'' follow directly from its existence when ''T'' is
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
. Then, just as in the bounded case, a complex number ''λ'' lies in the spectrum of a closed operator ''T'' if and only if T-\lambda I 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 Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
, then \sigma(T)=\Complex.


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. \, Tx\, \geq c\, x\, , for some c > 0, and has dense range. Accordingly, the spectrum of ''T'' can be divided into the following parts: # \lambda\in\sigma(T) if T - \lambda I is not bounded below. In particular, this is the case if T - \lambda I 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, T-\lambda I 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''). # \lambda\in\sigma(T) if T-\lambda I does not have dense range. The set of such ''λ'' is called the compression spectrum of ''T'', denoted by \sigma_(T). If T-\lambda I does not have dense range but is injective, ''λ'' is said to be in the residual spectrum of ''T'', denoted by \sigma_(T). 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 b ...
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 :\lim_ \, Tx_n - \lambda x_n\, = 0. The set of approximate eigenvalues is known as the approximate point spectrum, denoted by \sigma_(T). It is easy to see that the eigenvalues lie in the approximate point spectrum. For example, consider the right shift ''R'' on l^2(\Z) defined by :R:\,e_j\mapsto e_,\quad j\in\Z, where \big(e_j\big)_ is the standard orthonormal basis in l^2(\Z). Direct calculation shows ''R'' has no eigenvalues, but every ''λ'' with , ''λ'', = 1 is an approximate eigenvalue; letting ''x''''n'' be the vector :\frac(\dots, 0, 1, \lambda^, \lambda^, \dots, \lambda^, 0, \dots) one can see that , , ''x''''n'', , = 1 for all ''n'', but :\, Rx_n - \lambda x_n\, = \sqrt \to 0. 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 (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix i ...
, a bounded operator on a Hilbert space H is normal if and only if it is equivalent (after identification of ''H'' with an L^2 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 T-\lambda I is injective and has dense range, but is not surjective, is called the continuous spectrum of ''T'', denoted by \sigma_(T). The continuous spectrum therefore consists of those approximate eigenvalues which are not eigenvalues and do not lie in the residual spectrum. That is, :\sigma_(T) = \sigma_(T) \setminus (\sigma_(T) \cup \sigma_(T)) . For example, A:\,l^2(\N)\to l^2(\N), e_j\mapsto e_j/j, j\in\N, is injective and has a dense range, yet \mathrm(A)\subsetneq l^2(\N). Indeed, if x = \sum_ c_j e_j\in l^2(\N) with c_j \in \Complex such that \sum_ , c_j, ^2 < \infty, one does not necessarily have \sum_ \left, j c_j\^2 < \infty, and then \sum_ j c_j e_j \notin l^2(\N).


Compression spectrum

The set of \lambda\in\Complex for which T-\lambda I does not have dense range is known as the compression spectrum of ''T'' and is denoted by \sigma_(T).


Residual spectrum

The set of \lambda\in\Complex for which T-\lambda I is injective but does not have dense range is known as the residual spectrum of ''T'' and is denoted by \sigma_(T): :\sigma_(T) = \sigma_(T) \setminus \sigma_(T). An operator may be injective, even bounded below, but still not invertible. The right shift on l^2(\mathbb), R:\,l^2(\mathbb)\to l^2(\mathbb), R:\,e_j\mapsto e_,\,j\in\N, 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'' mea ...
, therefore bounded below by 1. But it is not invertible as it is not surjective (e_1\not\in\mathrm(R)), and moreover \mathrm(R) is not dense in l^2(\mathbb) (e_1\notin\overline).


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 A : \,X \to X which satisfy : \sigma_(A) \subset \sigma_(A) \subset \sigma_(A) \subset \sigma_(A) \subset \sigma_(A) \subset \sigma(A). All these spectra \sigma_(A),\ 1\le k\le 5, coincide in the case of self-adjoint operators. # The essential spectrum \sigma_(A) is defined as the set of points \lambda of the spectrum such that A-\lambda I 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: \lambda=0\in\sigma_(A) for the operator A:\,l^2(\N)\to l^2(\N), A:\,e_j\mapsto e_j/j,~ j\in\N (because the range of this operator is not closed: the range does not include all of l^2(\N) although its closure does).
Example 2: \lambda=0\in\sigma_(N) for N:\,l^2(\N)\to l^2(\N), N:\,v\mapsto 0 for any v\in l^2(\N) (because both kernel and cokernel of this operator are infinite-dimensional). # The essential spectrum \sigma_(A) is defined as the set of points \lambda of the spectrum such that the operator either A-\lambda I 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 ...
(x_j)_ in the space ''X'' such that \Vert x_j\Vert=1, \lim_ \left\, (A-\lambda I)x_j \right\, = 0, and such that (x_j)_ 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: \lambda=0\in\sigma_(B) for the operator B:\,l^2(\N)\to l^2(\N), B:\,e_j\mapsto e_ if ''j'' is even and e_j\mapsto 0 when ''j'' is odd (kernel is infinite-dimensional; cokernel is zero-dimensional). Note that \lambda=0\not\in\sigma_(B). # The essential spectrum \sigma_(A) is defined as the set of points \lambda of the spectrum such that A-\lambda I is not Fredholm. (The operator is ''Fredholm'' if its range is closed and both its kernel and cokernel are finite-dimensional.)
Example: \lambda=0\in\sigma_(J) for the operator J:\,l^2(\N)\to l^2(\N), J:\,e_j\mapsto e_ (kernel is zero-dimensional, cokernel is infinite-dimensional). Note that \lambda=0\not\in\sigma_(J). # The essential spectrum \sigma_(A) is defined as the set of points \lambda of the spectrum such that A-\lambda I 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, \sigma_(A) = \bigcap_ \sigma(A+K); here B_0(X) denotes the set of all compact operators on ''X''.
Example: \lambda=0\in\sigma_(R) where R:\,l^2(\N)\to l^2(\N) is the right shift operator, R:\,l^2(\N)\to l^2(\N), R:\,e_j\mapsto e_ for j\in\N (its kernel is zero, its cokernel is one-dimensional). Note that \lambda=0\not\in\sigma_(R). # The essential spectrum \sigma_(A) is the union of \sigma_(A) with all components of \Complex \setminus \sigma_(A) that do not intersect with the resolvent set \Complex \setminus \sigma(A). It can also be characterized as \sigma(A)\setminus\sigma_(A).
Example: consider the operator T:\,l^2(\Z)\to l^2(\Z), T:\,e_j\mapsto e_ for j\ne 0, T:\,e_0\mapsto 0. Since \Vert T\Vert=1, one has \sigma(T)\subset\overline. For any z\in\Complex with , z, =1, the range of T-z I is dense but not closed, hence the boundary of the unit disc is in the first type of the essential spectrum: \partial\mathbb_1\subset\sigma_(T). For any z\in\Complex with , z, <1, T-z I has a closed range, one-dimensional kernel, and one-dimensional cokernel, so z\in\sigma(T) although z\not\in\sigma_(T) for 1\le k\le 4; thus, \sigma_(T)=\partial\mathbb_1 for 1\le k\le 4. There are two components of \Complex\setminus\sigma_(T): \ and \. The component \ has no intersection with the resolvent set; by definition, \sigma_(T)=\sigma_(T)\cup\=\.


Example: Hydrogen atom

The
hydrogen atom A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen consti ...
provides an example of different types of the spectra. The hydrogen atom Hamiltonian operator H=-\Delta-\frac, Z > 0, with domain D(H) = H^1(\R^3) has a discrete set of eigenvalues (the discrete spectrum \sigma_(H), which in this case coincides with the point spectrum \sigma_(H) 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 *B ...
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 molecule i ...
process is described by the continuous part of the spectrum (the energy of the collision/ionization is not "quantized"), represented by \sigma_(H)=
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 ...
with dense domain D(T)\subset X. If ''X*'' is the dual space of ''X'', and T^*:\, X^* \to X^* is the hermitian adjoint of ''T'', then :\sigma(T^*) = \overline := \. We also get \sigma_(T)\subset\overline by the following argument: ''X'' embeds isometrically into ''X**''. Therefore, for every non-zero element in the kernel of T-\lambda I there exists a non-zero element in ''X**'' which vanishes on \mathrm(T^* - \barI). Thus \mathrm(T^* -\bar I) can not be dense. Furthermore, if ''X'' is reflexive, we have \overline\subset\sigma_(T).


Spectra of particular classes of operators


Compact operators

If ''T'' is a
compact operator In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact ...
, or, more generally, an inessential operator, then it can be shown that the spectrum is countable, that zero is the only possible
accumulation point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
, and that any nonzero ''λ'' in the spectrum is an eigenvalue.


Quasinilpotent operators

A bounded operator A:\,X\to X is quasinilpotent if \lVert A^n\rVert^ \to 0 as n\to\infty (in other words, if the spectral radius of ''A'' equals zero). Such operators could equivalently be characterized by the condition :\sigma(A)=\. An example of such an operator is A:\,l^2(\N)\to l^2(\N), e_j\mapsto e_/2^j for j\in\N.


Self-adjoint operators

If ''X'' is a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
and ''T'' is a
self-adjoint operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itse ...
(or, more generally, a
normal operator In mathematics, especially functional analysis, a normal operator on a complex Hilbert space ''H'' is a continuous linear operator ''N'' : ''H'' → ''H'' that commutes with its hermitian adjoint ''N*'', that is: ''NN*'' = ''N*N''. Normal operat ...
), then a remarkable result known as the
spectral theorem In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix i ...
gives an analogue of the diagonalisation theorem for normal finite-dimensional operators (Hermitian matrices, for example). For self-adjoint operators, one can use
spectral measure In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his disse ...
s to define a decomposition of the spectrum into absolutely continuous, pure point, and singular parts.


Spectrum of a real operator

The definitions of the resolvent and spectrum can be extended to any continuous linear operator T acting on a Banach space X over the real field \mathbb (instead of the complex field \mathbb) via its
complexification In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include t ...
T_\mathbb. In this case we define the resolvent set \rho(T) as the set of all \lambda\in\mathbb such that T_\mathbb-\lambda I is invertible as an operator acting on the complexified space X_\mathbb; then we define \sigma(T)=\mathbb\setminus\rho(T).


Real spectrum

The ''real spectrum'' of a continuous linear operator T acting on a real Banach space X, denoted \sigma_\mathbb(T), is defined as the set of all \lambda\in\mathbb for which T-\lambda I fails to be invertible in the real algebra of bounded linear operators acting on X. In this case we have \sigma(T)\cap\mathbb=\sigma_\mathbb(T). Note that the real spectrum may or may not coincide with the complex spectrum. In particular, the real spectrum could be empty.


Spectrum of a unital Banach algebra

Let ''B'' be a complex
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 spa ...
containing a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (alb ...
''e''. Then we define the spectrum ''σ''(''x'') (or more explicitly ''σ''''B''(''x'')) of an element ''x'' of ''B'' to be the set of those
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 form ...
s ''λ'' for which ''λe'' − ''x'' is not invertible in ''B''. This extends the definition for bounded linear operators ''B''(''X'') on a Banach space ''X'', since ''B''(''X'') is a Banach algebra.


See also

*
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 ...
*
Discrete spectrum (mathematics) In mathematics, specifically in spectral theory, a discrete spectrum of a closed linear operator is defined as the set of isolated points of its spectrum such that the rank (linear algebra), rank of the corresponding Riesz projector is finite. Def ...
*
Self-adjoint operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itse ...
*
Pseudospectrum In mathematics, the pseudospectrum of an operator is a set containing the spectrum of the operator and the numbers that are "almost" eigenvalues. Knowledge of the pseudospectrum can be particularly useful for understanding non-normal operators a ...
*
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 ...


References

* Dales et al., ''Introduction to Banach Algebras, Operators, and Harmonic Analysis'', * {{DEFAULTSORT:Spectrum (Functional Analysis) Spectral theory