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Encyclopedia > Spectrum of an operator

In functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of eigenvalues, which is much more 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 mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... 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 complex numbers are an extension of the real numbers by the inclusion of the imaginary unit , 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. ...

Contents

Definition

Let be a complex Banach space, and the Banach algebra of bounded linear operators on . Then if I denotes the identity operator, and T ∈ then the spectrum of T (normally written as σ(T) ) consists of λ such that λ I - T is not invertible in the algebra of bounded linear operators on . Note that by the closed graph theorem, this condition is equivalent to asserting λ I - T fails to be bijective. 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, 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. ...


Basic properties

Theorem: The spectrum is non-empty, bounded, and closed. In topology and related branches of mathematics, a closed set is a set whose complement is open. ...


Proof: Suppose the spectrum is empty; then the function R(λ) = (λI - T)-1 is defined everywhere on the complex plane. So if Φ is any linear functional on , F(λ) = Φ(R(λ)) is a continuous function CC. It is not hard to see that

so F is an analytic function. However, F(λ) is O-1) for large λ so F is a bounded analytic function, and hence constant by Liouville's theorem, and thus everywhere zero as it is zero at infinity. However, by the Hahn-Banach theorem this implies that R(λ) is zero for all λ, which is obviously a contradiction. In mathematics, an analytic function is one that is locally given by a convergent power series. ... Liouvilles theorem has various meanings: In complex analysis, see Liouvilles theorem (complex analysis). ... In mathematics, the Hahn-Banach theorem is a central tool in functional analysis. ...


The boundedness of the spectrum is immediate from the Neumann series expansion (named after the German mathematician Carl Neumann), Carl Gottfried Neumann was a German Mathematician, born May 7, 1832 in Königsberg (now Kaliningrad, Russia) and died March 27, 1925 in Leipzig. ...

,

which is valid for any A ∈ with ||A|| < 1. This implies that if |λ| > ||T||, (λ I - T) is invertible (taking A = T/λ). So σ(T) is bounded, and the spectral radius

is bounded above by ||T||.


Furthermore, the Neumann series implies that for any two operators A, B with A invertible and ||A - B|| < ||A-1||-1, B must also be invertible. It follows that the set of invertible operators is open, and hence, since the function C → defined by λ → λ I - T is continuous, the set of λ for which λ I - T is invertible is open, so its complement is closed; but this complement is exactly σ(T).


Classification of points in the spectrum

Loosely speaking, there are a variety of ways in which an operator S 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 S(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, S is not invertible if it is not bounded below; that is, if there is no 'c' > 0 such that ||Sx|| > c||x|| for all 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

The spectral radius formula states that

.

This can be proved using similar methods to the above theorem, considering the power series expansion of F(1/λ); this must converge for all λ > r(T), and applying the uniform boundedness principle to the series coefficients gives the result.


If T is a compact operator, then it can be shown that any nonzero approximate eigenvalue is in fact an eigenvalue. In functional analysis, a compact 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 bounded operator, and so...


If 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. ...


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. ...

External link

An account of the spectral theorem (http://www.srcf.ucam.org/~dl267/writeups/spectral_measures.pdf)


 
 

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