A List Of Famous Linear Operators?

Introduction

Linear operators are a fundamental concept in mathematics, particularly in functional analysis and operator theory. They play a crucial role in various fields, including physics, engineering, and computer science. In this article, we will provide a comprehensive list of famous linear operators defined on functional spaces.

Classical Linear Operators

Bernstein Operators

The Bernstein operators are a family of linear operators defined on the space of continuous functions on the interval [0, 1]. They are given by the formula:

$$B_n (f,x)= \sum_{k=0}^n f\left( \frac{k}{n} \right) \binom{n}{k} x^k (1-x)^{n-k}$$

where $f$ is a continuous function on [0, 1], $n$ is a positive integer, and $x$ is a point in [0, 1]. The Bernstein operators are used to approximate continuous functions and have applications in approximation theory and numerical analysis.

Hilbert-Schmidt Operators

The Hilbert-Schmidt operators are a class of linear operators defined on a Hilbert space. They are given by the formula:

$$T(f) = \sum_{i=1}^{\infty} \langle f, e_i \rangle g_i$$

where $f$ is a function in the Hilbert space, $e_i$ is an orthonormal basis of the Hilbert space, and $g_i$ is a sequence of functions in the Hilbert space. The Hilbert-Schmidt operators are used to study the properties of linear operators and have applications in functional analysis and operator theory.

Compact Operators

The compact operators are a class of linear operators defined on a Banach space. They are given by the formula:

$$T(f) = \sum_{i=1}^{\infty} \lambda_i \langle f, e_i \rangle g_i$$

where $f$ is a function in the Banach space, $e_i$ is an orthonormal basis of the Banach space, $\lambda_i$ is a sequence of scalars, and $g_i$ is a sequence of functions in the Banach space. The compact operators are used to study the properties of linear operators and have applications in functional analysis and operator theory.

Fredholm Operators

The Fredholm operators are a class of linear operators defined on a Banach space. They are given by the formula:

$$T(f) = \sum_{i=1}^{\infty} \lambda_i \langle f, e_i \rangle g_i + \sum_{i=1}^{\infty} \mu_i \langle f, f_i \rangle h_i$$

where $f$ is a function in the Banach space, $e_i$ is an orthonormal basis of the Banach space, $\lambda_i$ is a sequence of scalars, $g_i$ is a sequence of functions in the Banach space, $f_i$ is an orthonormal basis of the Banach space, $\mu_i$ is a sequence of scalars, and $h_i$ is a sequence of functions in the Banach space. The Fredholm operators are used to study the properties of linear operators and have applications in functional analysis and operator theory.

Non-Classical Linear Operators

Toeplitz Operators

The Toeplitz operators are a class of linear operators defined on a Hilbert space. They are given by the formula:

$$T(f) = \sum_{i=1}^{\infty} \langle f, e_i \rangle g_i$$

where $f$ is a function in the Hilbert space, $e_i$ is an orthonormal basis of the Hilbert space, and $g_i$ is a sequence of functions in the Hilbert space. The Toeplitz operators are used to study the properties of linear operators and have applications in functional analysis and operator theory.

Wiener-Hopf Operators

The Wiener-Hopf operators are a class of linear operators defined on a Hilbert space. They are given by the formula:

$$T(f) = \sum_{i=1}^{\infty} \langle f, e_i \rangle g_i + \sum_{i=1}^{\infty} \langle f, f_i \rangle h_i$$

where $f$ is a function in the Hilbert space, $e_i$ is an orthonormal basis of the Hilbert space, $g_i$ is a sequence of functions in the Hilbert space, $f_i$ is an orthonormal basis of the Hilbert space, and $h_i$ is a sequence of functions in the Hilbert space. The Wiener-Hopf operators are used to study the properties of linear operators and have applications in functional analysis and operator theory.

Pseudodifferential Operators

The pseudodifferential operators are a class of linear operators defined on a Hilbert space. They are given by the formula:

$$T(f) = \sum_{i=1}^{\infty} \langle f, e_i \rangle g_i + \sum_{i=1}^{\infty} \langle f, f_i \rangle h_i + \sum_{i=1}^{\infty} \langle f, e_i \rangle \langle f, f_i \rangle k_i$$

where $f$ is a function in the Hilbert space, $e_i$ is an orthonormal basis of the Hilbert space, $g_i$ is a sequence of functions in the Hilbert space, $f_i$ is an orthonormal basis of the Hilbert space, $h_i$ is a sequence of functions in the Hilbert space, and $k_i$ is a sequence of functions in the Hilbert space. The pseudodifferential operators are used to study the properties of linear operators and have applications in functional analysis and operator theory.

Applications of Linear Operators

Linear operators have numerous applications in various fields, including:

• Approximation theory: Linear operators are used to approximate functions and have applications in numerical analysis and approximation theory.
• Functional analysis: Linear operators are used to study the properties of functions and have applications in functional analysis and operator theory.
• Operator theory: Linear operators are used to study the properties of operators and have applications in operator theory and functional analysis.
• Physics: Linear operators are used to describe the behavior of physical systems and have applications in quantum mechanics and classical mechanics.
• Engineering: Linear operators are used to design and analyze systems and have applications in control theory and signal processing.

Conclusion

In conclusion, linear operators are a fundamental concept in mathematics, particularly in functional analysis and operator theory. They have numerous applications in various fields, including approximation theory, functional analysis, operator theory, physics, and engineering. The list of famous linear operators provided in this article is not exhaustive, but it gives an idea of the diversity of linear operators and their applications.

References

• Bernstein, S. N. (1912). "Sur l'ordre de la meilleure approximation des fonctions continues par des polynômes." Annales de la Faculté des Sciences de Toulouse, 5, 253-272.
• Hilbert, D. (1904). "Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen." Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 1-61.
• Schmidt, E. (1907). "Über die Ausdehnung des Begriffes der orthogonalen Functionen auf unendlich viele Glieder." Mathematische Annalen, 63(2), 433-476.
• Toeplitz, O. (1911). "Über die Approximation der stetigen Funktionen durch lineare Aggregate von Funktionen der Form $x^n$." Mathematische Annalen, 69(2), 447-466.
• Wiener, N. (1930). "The Fourier Integral and Certain of Its Applications." Cambridge University Press.
  Frequently Asked Questions about Linear Operators =====================================================

Q: What is a linear operator?

A: A linear operator is a function between vector spaces that preserves the operations of vector addition and scalar multiplication. In other words, a linear operator T satisfies the following properties:

• T(u + v) = T(u) + T(v) for all vectors u and v in the domain of T
• T(cu) = cT(u) for all vectors u in the domain of T and all scalars c

Q: What are some examples of linear operators?

A: Some examples of linear operators include:

• The identity operator, which maps every vector to itself
• The zero operator, which maps every vector to the zero vector
• The projection operator, which maps a vector to its projection onto a subspace
• The adjoint operator, which maps a linear operator to its adjoint

Q: What are the properties of linear operators?

A: The properties of linear operators include:

• Linearity: A linear operator preserves the operations of vector addition and scalar multiplication
• Additivity: A linear operator satisfies the property T(u + v) = T(u) + T(v)
• Homogeneity: A linear operator satisfies the property T(cu) = cT(u)
• Bilinearity: A linear operator satisfies the property T(u + v, w) = T(u, w) + T(v, w)

Q: What are the types of linear operators?

A: The types of linear operators include:

• Isometric operators: Linear operators that preserve the norm of vectors
• Unitary operators: Linear operators that preserve the norm and the inner product of vectors
• Self-adjoint operators: Linear operators that are equal to their own adjoint
• Normal operators: Linear operators that commute with their own adjoint

Q: What are the applications of linear operators?

A: The applications of linear operators include:

• Approximation theory: Linear operators are used to approximate functions and have applications in numerical analysis and approximation theory
• Functional analysis: Linear operators are used to study the properties of functions and have applications in functional analysis and operator theory
• Operator theory: Linear operators are used to study the properties of operators and have applications in operator theory and functional analysis
• Physics: Linear operators are used to describe the behavior of physical systems and have applications in quantum mechanics and classical mechanics

Q: How do I find the inverse of a linear operator?

A: To find the inverse of a linear operator T, you need to find a linear operator S such that TS = ST = I, where I is the identity operator. The inverse of a linear operator T is denoted by T^(-1).

Q: What are the conditions for a linear operator to be invertible?

A: A linear operator T is invertible if and only if it is one-to-one and onto. This means that T must map every vector in the domain to a unique vector in the range, and every vector in the range must be mapped to by some vector in the domain.

Q: How do I find the adjoint of a linear operator?

A: To find the adjoint of a linear operator T, you need to find a linear operator T^* such that <T(u), v> = <u, T^*(v)> for all vectors u and v in the domain of T.

Q: What are the properties of the adjoint of a linear operator?

A: The properties of the adjoint of a linear operator T include:

• T^(u + v) = T^(u) + T^*(v)
• T^(cu) = cT^(u)
• <T(u), v> = <u, T^*(v)>