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 is said to be in the spectrum of a bounded linear operator if
either has no set-theoretic inverse; or the set-theoretic inverse is either unbounded or defined on a non-dense subset.
Here, is the identity operator.
By the closed graph theorem, is in the spectrum if and only if the bounded operator is non-bijective on .
The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics.
The spectrum of an operator on a finite-dimensional vector space 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 x1=0, x2=0, etc. On the other hand, 0 is in the spectrum because although the operator R − 0 (i.e. R itself) is invertible, the inverse is defined on a set which is not dense in ℓ2. In fact every bounded linear operator on a complex Banach space must have a non-empty spectrum.
The notion of spectrum extends to unbounded (i.e. not necessarily bounded) operators. A complex number 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. 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 acting on a Banach space over the complex scalar field , and be the identity operator 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, it is bounded. Therefore, the spectrum consists precisely of those scalars for which is not bijective.
The spectrum of a given operator is often denoted , and its complement, the resolvent set, 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 is always a closed, bounded 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 is holomorphic 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 ; the spectrum is bounded by . A similar result shows the closedness of the spectrum.
The bound on the spectrum can be refined somewhat. The spectral radius, , of is the radius of the smallest circle in the complex plane which is centered at the origin and contains the spectrum inside of it, i.e.
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 of a Banach algebra,
Spectrum of an unbounded operator
One can extend the definition of spectrum to unbounded operators on a Banach space X. These operators 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, 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 .
The following example indicates that non-closed operators may have empty spectra. Let denote the differentiation operator on , whose domain is defined to be the closure of with respect to the -Sobolev space norm. This space can be characterized as all functions in that are zero at . Then, has trivial kernel on this domain, as any -function in its kernel is a constant multiple of , which is zero at if and only if it is identically zero. Therefore, the complement of the spectrum is all of
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 σp(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 σap(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 of T, one necessarily has λ ∈ σ(T). The set of eigenvalues of T is also called the point spectrum of T, denoted by σp(T). Some authors refer to the closure of the point spectrum as the pure point spectrum while others simply consider
Approximate point spectrum
More generally, by the bounded inverse theorem, 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 x1, x2, ... 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 bilateral shift W on defined by
where is the standard orthonormal basis in . Direct calculation shows W has no eigenvalues, but every λ with is an approximate eigenvalue; letting xn be the vector
one can see that ||xn|| = 1 for all n, but
Since W is a unitary operator, its spectrum lies on the unit circle. Therefore, the approximate point spectrum of W is its entire spectrum.
This conclusion is also true for a more general class of operators. A unitary operator is normal. By the spectral theorem, 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. It can be shown that the approximate point spectrum of a bounded multiplication operator equals its spectrum.
Discrete spectrum
The discrete spectrum is defined as the set of normal eigenvalues or, equivalently, as the set of isolated points of the spectrum such that the corresponding Riesz projector is of finite rank. As such, the discrete spectrum is a strict subset of the point spectrum, i.e.,
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, 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.
Essential spectrum
There are five similar definitions of the essential spectrum 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).