Operator ideal

Last updated November 02, 2019

In functional analysis, a branch of mathematics, an operator ideal is a special kind of class of continuous linear operators between Banach spaces. If an operator $T$ belongs to an operator ideal ${\mathcal {J}}$ , then for any operators $A$ and $B$ which can be composed with $T$ as $BTA$ , then $BTA$ is class ${\mathcal {J}}$ as well. Additionally, in order for ${\mathcal {J}}$ to be an operator ideal, it must contain the class of all finite-rank Banach space operators.

Functional analysis branch of mathematical analysis concerned with infinite-dimensional topological vector spaces, often spaces of functions

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

In set theory and its applications throughout mathematics, a class is a collection of sets that can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context. In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity.

Formal definition

Let ${\mathcal {L}}$ denote the class of continuous linear operators acting between arbitrary Banach spaces. For any subclass ${\mathcal {J}}$ of ${\mathcal {L}}$ and any two Banach spaces $X$ and $Y$ over the same field $\mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}$ , denote by ${\mathcal {J}}(X,Y)$ the set of continuous linear operators of the form $T:X\to Y$ such that $T\in {\mathcal {J}}$ . In this case, we say that ${\mathcal {J}}(X,Y)$ is a component of ${\mathcal {J}}$ . An operator ideal is a subclass ${\mathcal {J}}$ of ${\mathcal {L}}$ , containing every identity operator acting on a 1-dimensional Banach space, such that for any two Banach spaces $X$ and $Y$ over the same field $\mathbb {K}$ , the following two conditions for ${\mathcal {J}}(X,Y)$ are satisfied:

(1) If

S,T\in {\mathcal {J}}(X,Y)

then

S+T\in {\mathcal {J}}(X,Y)

; and

(2) if

W

and

Z

are Banach spaces over

\mathbb {K}

with

A\in {\mathcal {L}}(W,X)

and

B\in {\mathcal {L}}(Y,Z)

, and if

T\in {\mathcal {J}}(X,Y)

, then

BTA\in {\mathcal {J}}(W,Z)

.

Properties and examples

Operator ideals enjoy the following nice properties.

Every component ${\mathcal {J}}(X,Y)$ of an operator ideal forms a linear subspace of ${\mathcal {L}}(X,Y)$ , although in general this need not be norm-closed.
Every operator ideal contains all finite-rank operators. In particular, the finite-rank operators form the smallest operator ideal.
For each operator ideal ${\mathcal {J}}$ , every component of the form ${\mathcal {J}}(X):={\mathcal {J}}(X,X)$ forms an ideal in the algebraic sense.

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any other integer results in another even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring similarly to the way that, in group theory, a normal subgroup can be used to construct a quotient group.

Furthermore, some very well-known classes are norm-closed operator ideals, i.e., operator ideals whose components are always norm-closed. These include but are not limited to the following.

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In functional analysis, a branch of mathematics, a strictly singular operator is a bounded linear operator between normed spaces which is not bounded below on any infinite-dimensional subspace.

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In mathematics, any vector space V has a corresponding dual vector space consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

In mathematics, a normed vector space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real world. A norm is a real-valued function defined on the vector space that has the following properties:

The zero vector, 0, has zero length; every other vector has a positive length.
Multiplying a vector by a positive number changes its length without changing its direction. Moreover,
The triangle inequality holds. That is, taking norms as distances, the distance from point A through B to C is never shorter than going directly from A to C, or the shortest distance between any two points is a straight line.

In mathematics, an operator is generally a mapping that acts on elements of a space to produce elements of another space. The most common operators are linear maps, which act on vector spaces. However, when using "linear operator" instead of "linear map", mathematicians often mean actions on vector spaces of functions, which also preserve other properties, such as continuity. For example, differentiation and indefinite integration are linear operators; operators that are built from them are called differential operators, integral operators or integro-differential operators.

In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence $of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.$

In algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by $, is the set of all prime ideals of R . It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. A locally ringed space of this form is called an affine scheme .$

In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if $is not invertible, where I is the identity operator. The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics.$

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In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.

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In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

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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, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products, but for general Banach spaces or locally convex topological vector spaces the theory is notoriously subtle.

In functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators in the topology induced by the operator norm. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. By contrast, the study of general operators on infinite-dimensional spaces often requires a genuinely different approach.

In functional analysis, the dual norm is a measure of the "size" of each continuous linear functional defined on a normed vector space.

References

Pietsch, Albrecht: Operator Ideals, Volume 16 of Mathematische Monographien, Deutscher Verlag d. Wiss., VEB, 1978.

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Operator ideal

Contents

Formal definition

Properties and examples

Related Research Articles

References