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In functional analysis, the concept of the spectrum of an element of a Banach algebra is a generalisation of the concept of eigenvalues, which is exceedingly useful in the case of operators on infinite-dimensional spaces. For example, the bilateral shift operator on the Hilbert space has no eigenvalues at all; but we shall see below that any bounded linear operator on a complex Banach space must have non-empty spectrum. Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ...
In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ...
In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...
In mathematics, and in particular functional analysis, the shift operators are examples of linear operators, important for their simplicity and natural occurrence. ...
In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ...
In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ...
In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...
In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...
The study of the properties of spectra 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. ...
Definition
Let B be a complex Banach algebra containing a unit e. The spectrum of an element x of B, often written as σB(x) or simply σ(x), consists of those complex numbers λ for which λ e - x is not invertible in B. In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ...
In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of...
In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...
Basic properties The spectrum σ(x) of an element x of B is always compact and non-empty. If the spectrum were empty, then the resolvent function In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ...
In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...
- R(λ) = (λe - x)-1
would be defined everywhere on the complex plane and bounded, which would imply by Liouville's theorem that this function is constant, thus everywhere zero as it is zero at infinity, which would be a contradiction. The boundedness of the spectrum follows from the Neumann series expansion in λ, which also helps prove the closedness of the spectrum and hence its compactness. Liouvilles theorem in complex analysis states that every bounded (i. ...
One defines the spectral radius to be the radius of the largest circle in the complex plane which is centered at the origin and contains the spectrum σ(x) inside of it: In mathematics, the spectral radius of a matrix or a bounded linear operator is the supremum among the moduli of the elements in its spectrum, which is sometimes denoted by ρ(·). Matrix Let λ1,...,λn be the (real or complex) eigenvalues of a matrix A. Then ρ(A) := ρ(A...
One can prove that r(x) is bounded above by ||x||. The spectral radius formula is a refinement of this statement. It says that
Spectrum of a bounded operator If X is a complex Banach space, then the set of all bounded linear operators on X forms a Banach algebra, denoted by B(X). This makes it possible to define the spectrum of a bounded linear operator viewed as an element in this Banach algebra. More specifically, denote by I the identity operator on X, so that I is the unit of B(X). Then for T ∈ B(X) a bounded linear operator, the spectrum of T, written σ(T), consists of those λ for which λ I - T is not invertible in B(X). Note that by the closed graph theorem, this condition is equivalent to asserting that λ I - T fails to be bijective. In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...
In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ...
In mathematics, the closed graph theorem is a basic result of functional analysis. ...
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...
Classification of points in the spectrum of an operator Loosely speaking, there are a variety of ways in which an operator T can fail to be invertible, and this allows us to classify the points of the spectrum into various types.
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, we necessarily have λ ∈ σ(T). The set of eigenvalues of T is sometimes called the point spectrum of T. In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...
Approximate point spectrum More generally, 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 x ∈ X. So the spectrum includes the set of approximate eigenvalues, which are those λ such that T - λ I 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.
For example, in the example in the first paragraph of the bilateral shift on , there are no eigenvectors, but every λ with |λ| = 1 is an approximate eigenvector; letting xn be the vector In mathematics, and in particular functional analysis, the shift operators are examples of linear operators, important for their simplicity and natural occurrence. ...
then ||xn|| = 1 for all n, but - .
Compression spectrum The unilateral shift on gives an example of yet another way in which an operator can fail to be invertible; this shift operator is bounded below (by 1; it is obviously norm-preserving) but it is not invertible as it is not surjective. The set of λ for which λ I - T is not surjective is known as the compression spectrum of T. In mathematics, and in particular functional analysis, the shift operators are examples of linear operators, important for their simplicity and natural occurrence. ...
This exhausts the possibilities, since if T is surjective and bounded below, T is invertible.
Further results If T is a compact operator, then it can be shown that any nonzero λ in the spectrum is an eigenvalue. In other words, the spectrum of such an operator, which was defined as a generalization of the concept of eigenvalues, consists in this case only of the usual eigenvalues, and possibly 0. In functional analysis, a compact operator (or completely continuous operator) is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an operator is necessarily a...
If X is a Hilbert space and T is a normal operator, then a remarkable result known as the spectral theorem gives an analogue of the diagonalisation theorem for normal finite-dimensional operators (Hermitian matrices, for example). In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ...
In functional analysis, a normal operator on a Hilbert space H is a continuous linear operator N : H → H that commutes with its hermitian adjoint N*: N N* = N* N. The main importance of this concept is that the spectral theorem applies to normal operators. ...
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ...
Spectrum of unbounded operators One can extend the definition of spectrum for unbounded operators on a Banach space X, operators which are no longer elements in the Banach algebra B(X). One proceeds in a manner similar to the bounded case. A complex number λ is said to be in the complement of the spectrum of a linear operator In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...
In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...
if the operator is surjective, with its inverse a bounded operator. A complex number λ is then in the spectrum if this property fails to hold. In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...
In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...
The spectrum of an unbounded operator is in general an unbounded subset of the complex plane.
See also In mathematics and physics, discrete spectrum is a finite set or a countable set of eigenvalues of an operator. ...
In mathematics and physics, continuous spectrum is, roughly speaking, a non-countable set of eigenvalues of an operator. ...
In mathematics, the essential spectrum of a bounded operator is a certain subset of its spectrum, defined by a condition of the type that says, roughly speaking, fails badly to be invertible. The essential spectrum of self-adjoint operators In formal terms, let X be a Hilbert space and let...
On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ...
References - Dales et al, Introduction to Banach Algebras, Operators, and Harmonic Analysis
External links - An account of the spectral theorem
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