Operator theory

In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis.

If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory.

Single operator theory

Single operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of normal operators in terms of their spectra falls into this category.

Spectrum of operators

Main article: Spectral theorem

The spectral theorem is any of a number of results about linear operators or about matrices.^[1] In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This concept of diagonalization is relatively straightforward for operators on finite-dimensional spaces, but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modelled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective.

Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces.

The spectral theorem also provides a canonical decomposition, called the spectral decomposition, eigenvalue decomposition, or eigendecomposition, of the underlying vector space on which the operator acts.

Normal operators

Main article: Normal operator

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.^[2]

Normal operators are important because the spectral theorem holds for them. Today, the class of normal operators is well-understood. Examples of normal operators are

unitary operators: N* = N⁻¹
Hermitian operators (i.e., selfadjoint operators): N* = N; (also, anti-selfadjoint operators: N* = −N)
positive operators: N = MM*
normal matrices can be seen as normal operators if one takes the Hilbert space to be Cⁿ.

The spectral theorem extends to a more general class of matrices. Let A be an operator on a finite-dimensional inner product space. A is said to be normal if A^* A = A A^*. One can show that A is normal if and only if it is unitarily diagonalizable: By the Schur decomposition, we have A = U T U^*, where U is unitary and T upper-triangular. Since A is normal, T T^* = T^* T. Therefore, T must be diagonal since normal upper triangular matrices are diagonal. The converse is obvious.

In other words, A is normal if and only if there exists a unitary matrix U such that

A=U D U^* \;

where D is a diagonal matrix. Then, the entries of the diagonal of D are the eigenvalues of A. The column vectors of U are the eigenvectors of A and they are orthonormal. Unlike the Hermitian case, the entries of D need not be real.

Polar decomposition

Main article: Polar decomposition

The polar decomposition of any bounded linear operator A between complex Hilbert spaces is a canonical factorization as the product of a partial isometry and a non-negative operator.^[3]

The polar decomposition for matrices generalizes as follows: if A is a bounded linear operator then there is a unique factorization of A as a product A = UP where U is a partial isometry, P is a non-negative self-adjoint operator and the initial space of U is the closure of the range of P.

The operator U must be weakened to a partial isometry, rather than unitary, because of the following issues. If A is the one-sided shift on l²(N), then |A| = {A*A}^½ = I. So if A = U |A|, U must be A, which is not unitary.

The existence of a polar decomposition is a consequence of Douglas' lemma:

Lemma If A, B are bounded operators on a Hilbert space H, and A*A ≤ B*B, then there exists a contraction C such that A = CB. Furthermore, C is unique if Ker(B*) ⊂ Ker(C).

The operator C can be defined by C(Bh) = Ah, extended by continuity to the closure of Ran(B), and by zero on the orthogonal complement of Ran(B). The operator C is well-defined since A*A ≤ B*B implies Ker(B) ⊂ Ker(A). The lemma then follows.

In particular, if A*A = B*B, then C is a partial isometry, which is unique if Ker(B*) ⊂ Ker(C). In general, for any bounded operator A,

A^*A = (A^*A)^{\frac{1}{2}} (A^*A)^{\frac{1}{2}},

where (A*A)^½ is the unique positive square root of A*A given by the usual functional calculus. So by the lemma, we have

A = U (A^*A)^{\frac{1}{2}}

for some partial isometry U, which is unique if Ker(A) ⊂ Ker(U). (Note Ker(A)=Ker(A*A)=Ker(B)=Ker(B*), where B=B*=(A*A)^½.) Take P to be (A*A)^½ and one obtains the polar decomposition A = UP. Notice that an analogous argument can be used to show A = P'U' , where P' is positive and U' a partial isometry.

When H is finite dimensional, U can be extended to a unitary operator; this is not true in general (see example above). Alternatively, the polar decomposition can be shown using the operator version of singular value decomposition.

By property of the continuous functional calculus, |A| is in the C*-algebra generated by A. A similar but weaker statement holds for the partial isometry: the polar part U is in the von Neumann algebra generated by A. If A is invertible, U will be in the C*-algebra generated by A as well.

Operator algebras

The theory of operator algebras brings algebras of operators such as C*-algebras to the fore.

C*-algebras

Main article: C*-algebra

A C*-algebra, A, is a Banach algebra over the field of complex numbers, together with a map * : A → A. One writes x* for the image of an element x of A. The map * has the following properties:^[4]

It is an involution, for every x in A

x^{**}=(x^{*})^{*}=x

For all x, y in A:

(x+y)^{*}=x^{*}+y^{*}

(xy)^{*}=y^{*}x^{*}

For every λ in C and every x in A:

(\lambda x)^{*}={\overline {\lambda }}x^{*}.

For all x in A:

\|x^{*}x\|=\|x\|\|x^{*}\|.

Remark. The first three identities say that A is a *-algebra. The last identity is called the C* identity and is equivalent to:

$\|xx^{*}\|=\|x\|^{2},$

The C*-identity is a very strong requirement. For instance, together with the spectral radius formula, it implies that the C*-norm is uniquely determined by the algebraic structure:

\|x\|^{2}=\|x^{*}x\|=\sup\{|\lambda |:x^{*}x-\lambda \,1{\text{ is not invertible}}\}.

References

↑ Sunder, V.S. Functional Analysis: Spectral Theory (1997) Birkhäuser Verlag
↑ Hoffman, Kenneth; Kunze, Ray (1971), Linear algebra (2nd ed.), Englewood Cliffs, N.J.: Prentice-Hall, Inc., p. 312, MR 0276251
↑ Conway, John B. (2000), A Course in Operator Theory, Graduate Studies in Mathematics, American Mathematical Society, ISBN 0821820656
↑ Arveson, W. (1976), An Invitation to C*-Algebra, Springer-Verlag, ISBN 0-387-90176-0 . An excellent introduction to the subject, accessible for those with a knowledge of basic functional analysis.

External links

History of Operator Theory

Functional analysis

Set / subset types	Absolutely convex Absorbing Balanced Bounded Convex Radial Star-shaped Symmetric Linear cone (subset) Convex cone (subset)

TVS types	Banach Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Webbed Topological tensor product (of Hilbert spaces)

Mapping topologies	Dual Dual space Operator Ultraweak Weak (operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence

Linear operators	Adjoint Bilinear (form) Bounded / Unbounded Closed Continuous / Discontinuous Compact Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary

Set operations	Algebraic interior (core) Interior Minkowski addition Polar

Operator theory	Banach algebras C-algebras Spectrum (C-algebra radius) Spectral theory (of ODEs Spectral theorem) Polar decomposition Singular value decomposition

Theorems	Banach–Alaoglu Banach–Saks Banach–Mazur Bessel's inequality Cauchy–Schwarz inequality Closed range Closed graph Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Open mapping Parseval's identity Schauder fixed-point

Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure Weakly measurable function

This article is issued from Wikipedia - version of the 10/14/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.