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In operator theory, a **Toeplitz operator** is the compression of a multiplication operator on the circle to the Hardy space.

Let *S*^{1} be the circle, with the standard Lebesgue measure, and *L*^{2}(*S*^{1}) be the Hilbert space of square-integrable functions. A bounded measurable function *g* on *S*^{1} defines a multiplication operator *M _{g}* on

where " | " means restriction.

A bounded operator on *H*^{2} is Toeplitz if and only if its matrix representation, in the basis {*z ^{n}*,

- Theorem: If is continuous, then is Fredholm if and only if is not in the set . If it is Fredholm, its index is minus the winding number of the curve traced out by with respect to the origin.

For a proof, see Douglas (1972 , p.185). He attributes the theorem to Mark Krein, Harold Widom, and Allen Devinatz. This can be thought of as an important special case of the Atiyah-Singer index theorem.

Here, denotes the closed subalgebra of of analytic functions (functions with vanishing negative Fourier coefficients), is the closed subalgebra of generated by and , and is the space (as an algebraic set) of continuous functions on the circle. See S.Axler, S-Y. Chang, D. Sarason (1978).

In mathematics, especially functional analysis, a **Banach algebra**, named after Stefan Banach, is an associative algebra over the real or complex numbers that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. The norm is required to satisfy

In mathematics, specifically in functional analysis, a **C ^{∗}-algebra** is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra

In mathematics, particularly linear algebra and functional analysis, a **spectral theorem** is a result about when a linear operator or matrix can be diagonalized. This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector 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 modeled 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.

In mathematics, a **self-adjoint operator** on an infinite-dimensional complex vector space *V* with inner product is a linear map *A* that is its own adjoint. If *V* is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of *A* is a Hermitian matrix, i.e., equal to its conjugate transpose *A*^{∗}. By the finite-dimensional spectral theorem, *V* has an orthonormal basis such that the matrix of *A* relative to this basis is a diagonal matrix with entries in the real numbers. In this article, we consider generalizations of this concept to operators on Hilbert spaces of arbitrary dimension.

In mathematics, **Fredholm operators** are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator *T* : *X* → *Y* between two Banach spaces with finite-dimensional kernel and finite-dimensional (algebraic) cokernel , and with closed range . The last condition is actually redundant.

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.

In mathematics, a Lie algebra is **semisimple** if it is a direct sum of simple Lie algebras..

In mathematics, the **Fredholm determinant** is a complex-valued function which generalizes the determinant of a finite dimensional linear operator. It is defined for bounded operators on a Hilbert space which differ from the identity operator by a trace-class operator. The function is named after the mathematician Erik Ivar Fredholm.

In mathematics, a **Fredholm kernel** is a certain type of a kernel on a Banach space, associated with nuclear operators on the Banach space. They are an abstraction of the idea of the Fredholm integral equation and the Fredholm operator, and are one of the objects of study in Fredholm theory. Fredholm kernels are named in honour of Erik Ivar Fredholm. Much of the abstract theory of Fredholm kernels was developed by Alexander Grothendieck and published in 1955.

In mathematics, **ergodic flows** occur in geometry, through the geodesic and horocycle flows of closed hyperbolic surfaces. Both of these examples have been understood in terms of the theory of unitary representations of locally compact groups: if Γ is the fundamental group of a closed surface, regarded as a discrete subgroup of the Möbius group G = PSL(2,**R**), then the geodesic and horocycle flow can be identified with the natural actions of the subgroups *A* of real positive diagonal matrices and *N* of lower unitriangular matrices on the unit tangent bundle *G* / Γ. The Ambrose-Kakutani theorem expresses every ergodic flow as the flow built from an invertible ergodic transformation on a measure space using a ceiling function. In the case of geodesic flow, the ergodic transformation can be understood in terms of symbolic dynamics; and in terms of the ergodic actions of Γ on the boundary *S*^{1} = *G* / *AN* and *G* / *A* = *S*^{1} × *S*^{1} \ diag *S*^{1}. Ergodic flows also arise naturally as invariants in the classification of von Neumann algebras: the flow of weights for a factor of type III_{0} is an ergodic flow on a measure space.

In the theory of von Neumann algebras, a part of the mathematical field of functional analysis, **Tomita–Takesaki theory** is a method for constructing **modular automorphisms** of von Neumann algebras from the polar decomposition of a certain involution. It is essential for the theory of type III factors, and has led to a good structure theory for these previously intractable objects.

In mathematics, the **Fredholm alternative**, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators. Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue.

In operator algebras, the **Toeplitz algebra** is the C*-algebra generated by the unilateral shift on the Hilbert space *l*^{2}(**N**). Taking *l*^{2}(**N**) to be the Hardy space *H*^{2}, the Toeplitz algebra consists of elements of the form

In complex analysis, given *initial data* consisting of points in the complex unit disc and *target data* consisting of points in , the **Nevanlinna–Pick interpolation problem** is to find a holomorphic function that interpolates the data, that is for all ,

In mathematics, the **Plancherel theorem for spherical functions** is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish-Chandra. It is a natural generalisation in non-commutative harmonic analysis of the Plancherel formula and Fourier inversion formula in the representation theory of the group of real numbers in classical harmonic analysis and has a similarly close interconnection with the theory of differential equations. It is the special case for zonal spherical functions of the general Plancherel theorem for semisimple Lie groups, also proved by Harish-Chandra. The Plancherel theorem gives the eigenfunction expansion of radial functions for the Laplacian operator on the associated symmetric space *X*; it also gives the direct integral decomposition into irreducible representations of the regular representation on *L*^{2}(*X*). In the case of hyperbolic space, these expansions were known from prior results of Mehler, Weyl and Fock.

In mathematics, a **commutation theorem for traces** explicitly identifies the commutant of a specific von Neumann algebra acting on a Hilbert space in the presence of a trace.

In mathematics, **Hilbert spaces** 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 naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space.

**Donald Erik Sarason** was an American mathematician who made fundamental advances in the areas of Hardy space theory and VMO. He was one of the most popular doctoral advisors in the Mathematics Department at UC Berkeley. He supervised 39 Ph.D. theses at UC Berkeley.

In functional analysis, every C^{*}-algebra is isomorphic to a subalgebra of the C^{*}-algebra of bounded linear operators on some Hilbert space This article describes the spectral theory of closed normal subalgebras of . A subalgebra of is called normal if it is commutative and closed under the operation: for all , we have and that .

This is a glossary for the terminology in a mathematical field of functional analysis.

- S.Axler, S-Y. Chang, D. Sarason (1978), "Products of Toeplitz operators",
*Integral Equations and Operator Theory*,**1**(3): 285–309, doi:10.1007/BF01682841, S2CID 120610368`{{citation}}`

: CS1 maint: multiple names: authors list (link) - Böttcher, Albrecht; Grudsky, Sergei M. (2000),
*Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis*, Birkhäuser, ISBN 978-3-0348-8395-5 . - Böttcher, A.; Silbermann, B. (2006),
*Analysis of Toeplitz Operators*, Springer Monographs in Mathematics (2nd ed.), Springer-Verlag, ISBN 978-3-540-32434-8 . - Douglas, Ronald (1972),
*Banach Algebra techniques in Operator theory*, Academic Press. - Rosenblum, Marvin; Rovnyak, James (1985),
*Hardy Classes and Operator Theory*, Oxford University Press. Reprinted by Dover Publications, 1997, ISBN 978-0-486-69536-5.

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