算子 | 类型 / 性质-优快云博客

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Operator - 算子

Definitions and Notations - 定义与符号

Domain and Range - 定义域和值域

The subset $D$ is called the domain of definition of the operator $A$ and is denoted by $\operatorname{Dom}(A)$ ; the set $\{A(x) : x \in D\}$ is called the domain of values of the operator $A$ (or its range) and is denoted by $\operatorname{R}(A)$ .
子集 $D$ 称为算子 $A$ 的定义域，记作 $\operatorname{Dom}(A)$ ；集合 $\{A(x) : x \in D\}$ 称为算子 $A$ 的值域，记作 $\operatorname{R}(A)$ 。

Operators on the Same Space - 同一空间上的算子

If $A$ is an operator from $X$ into $Y$ where $X = Y$ , then $A$ is called an operator on $X$ .
如果 $A$ 是从 $X$ 到 $Y$ 的算子，且 $X = Y$ ，则称 $A$ 为 $X$ 上的算子。

Everywhere-Defined Operator - 处处定义的算

If $\operatorname{Dom}(A) = X$ , then $A$ is called an everywhere-defined operator.
若 $\operatorname{Dom}(A) = X$ ，则称 $A$ 为处处定义的算子。

Restriction and Extension - 限制与扩张

If $A_1$ , $A_2$ are operators from $X_1$ into $Y_1$ and from $X_2$ into $Y_2$ with domains of definition $\operatorname{Dom}(A_1)$ and $\operatorname{Dom}(A_2)$ , respectively, such that $\operatorname{Dom}(A_1) \subset \operatorname{Dom}(A_2)$ and $A_1x = A_2x$ for all $\in \operatorname{Dom}(A_1)$ , then:
设 $A_1$ 和 $A_2$ 分别是从 $X_1$ 到 $Y_1$ 和从 $X_2$ 到 $Y_2$ 的算子，定义域分别为 $\operatorname{Dom}(A_1)$ 和 $\operatorname{Dom}(A_2)$ 。若 $\operatorname{Dom}(A_1) \subset \operatorname{Dom}(A_2)$ 且对所有 $\in \operatorname{Dom}(A_1)$ 有 $A_1x = A_2x$ ，则：
if $X_1 = X_2$ , $Y_1 = Y_2$ , the operator $A_1$ is called a compression or restriction of the operator $A_2$ , while $A_2$ is called an extension of $A_1$ ;
当 $X_1 = X_2$ 且 $Y_1 = Y_2$ 时，称 $A_1$ 是 $A_2$ 的压缩或限制， $A_2$ 是 $A_1$ 的扩张；
if $X_1 \subset X_2$ , $A_2$ is called an extension of $A_1$ exceeding $X_1$ .
当 $X_1 \subset X_2$ 时，称 $A_2$ 是 $A_1$ 的扩张，且超出 $X_1$ 。

Linear and Non-linear Operators - 线性与非线性算子

If $X$ and $Y$ are vector spaces, then in the set of all operators from $X$ into $Y$ , the class of linear operators can be singled out; the remaining operators are called non-linear operators.
若 $X$ 和 $Y$ 是向量空间，则在所有从 $X$ 到 $Y$ 的算子中，可区分出线性算子类；其余算子称为非线性算子。

Continuous, Bounded, and Compact Operators - 连续、有界与紧算子

If $X$ and $Y$ are topological vector spaces, the set of operators from $X$ into $Y$ includes:
若 $X$ 和 $Y$ 是拓扑向量空间，从 $X$ 到 $Y$ 的算子包括：
Continuous operators: naturally defined by continuous mappings;
连续算子：由连续映射自然定义；
Bounded linear operators: mappings where any bounded set in $X$ has a bounded image in $Y$ ;
有界线性算子： $X$ 中任意有界集的像在 $Y$ 中有界；
Compact linear operators: mappings where any bounded set in $X$ has a pre-compact image in $Y$ .
紧线性算子： $X$ 中任意有界集的像在 $Y$ 中是预紧的。

Semi-Continuous, Strongly Continuous, and Weakly Continuous Operators - 半连续、强连续与弱连续算子

If $X$ and $Y$ are locally convex spaces:
若 $X$ 和 $Y$ 是局部凸空间：
An operator is semi-continuous if it maps $X$ (with the initial topology) continuously into $Y$ (with the weak topology), mainly used in non-linear operator theory;
若算子从 $X$ （初始拓扑）到 $Y$ （弱拓扑）连续，则为半连续算子（主要用于非线性算子理论）；
An operator is strongly continuous if it is continuous from $X$ (with the boundedly weak topology) to $Y$ ;
若算子从 $X$ （有界弱拓扑）到 $Y$ 连续，则为强连续算子；
An operator is weakly continuous if it is continuous between $X$ and $Y$ with both having the weak topology.
若算子在 $X$ 和 $Y$ 的弱拓扑下连续，则为弱连续算子。

Closed Operators - 闭算子

If $X$ and $Y$ are topological vector spaces, an operator from $X$ into $Y$ is closed if its graph $\Gamma(A) = \{(x, Ax) : x \in \operatorname{Dom}(A)\}$ is closed. This concept is particularly useful for linear operators with a dense domain.
若 $X$ 和 $Y$ 是拓扑向量空间，从 $X$ 到 $Y$ 的算子若其图像 $\Gamma(A) = \{(x, Ax) : x \in \operatorname{Dom}(A)\}$ 是闭集，则称为闭算子。该概念对定义域稠密的线性算子尤为重要。

Connection with Equations - 与方程的联系

Many equations in function spaces can be written as $A x = y$ , where $A$ is an operator from $X$ to $Y$ , $\in Y$ is given, and $\in X$ is unknown:
函数空间中许多方程可表示为 $A x = y$ ，其中 $A$ 是从 $X$ 到 $Y$ 的算子， $\in Y$ 已知， $\in X$ 未知：

Existence of a solution for any $\in Y$ is equivalent to $\operatorname{R}(A) = Y$ ;
对任意 $\in Y$ 有解，等价于 $\operatorname{R}(A) = Y$ ；
Unique solution for any $\in \operatorname{R}(A)$ means $A$ is a bijection from $\operatorname{Dom}(A)$ to $\operatorname{R}(A)$ .
对任意 $\in \operatorname{R}(A)$ 有唯一解，意味着 $A$ 是 $\operatorname{Dom}(A)$ 到 $\operatorname{R}(A)$ 的双射。

Examples of Operators - 算子的例子

1. Zero Operator - 零算子

Maps any $\in X$ to $\in Y$ , denoted $0$ .
将任意 $\in X$ 映射为 $\in Y$ ，记作 $0$ 。

2. Identity Operator - 恒等算子

Maps $\in X$ to itself, denoted $\mathop{\rm id}\nolimits_X$ or $1_X$ .
将 $\in X$ 映射为自身，记作 $\mathop{\rm id}\nolimits_X$ 或 $1_X$ 。

3. Multiplication Operator - 乘法算子

Let $X$ be a function space on set $M$ , $\in X$ . The operator $A$ with:
设 $X$ 是集合 $M$ 上的函数空间， $\in X$ 。算子 $A$ 定义为：
$\{ \phi \in X : f\phi \in X \}, \quad A\phi = f\phi,$
is a linear operator called multiplication by a function.
称为函数乘法算子，是线性算子。

4. Composition Operator - 复合算子

Let $X$ be a function space on set $M$ , $\to M$ . The operator $A$ with:
设 $X$ 是集合 $M$ 上的函数空间， $\to M$ 。算子 $A$ 定义为：
$\{ \phi \in X : \phi \circ F \in X \}, \quad A\phi = \phi \circ F,$
is a linear operator called composition operator.
称为复合算子，是线性算子。

5. Integral Operator - 积分算子

Let $\subset L^2(M, \mu)$ , $\subset L^2(N, \nu)$ , and $K (x, y, z)$ be a measurable function. The operator $A$ with:
设 $\subset L^2(M, \mu)$ ， $\subset L^2(N, \nu)$ ， $K (x, y, z)$ 为可测函数。算子 $A$ 定义为：
$\left\{ \phi \in X : f(x) = \int_M K(x, y, \phi(y)) \, dy \in Y \right\}, \quad A\phi = f,$
is an integral operator. If $K (x, y, z) = K (x, y) z$ , $A$ is linear.
称为积分算子。若 $K (x, y, z) = K (x, y) z$ ，则 $A$ 是线性的。

6. Differentiation Operator - 微分算子

Let $X$ be a function space on a differentiable manifold $M$ , $\xi$ a vector field on $M$ . The operator $A$ with:
设 $X$ 是可微流形 $M$ 上的函数空间， $\xi$ 是 $M$ 上的向量场。算子 $A$ 定义为：
$\{ f \in X : D_\xi f \text{ exists and } D_\xi f \in X \}, \quad Af = D_\xi f,$
is a linear operator called differentiation operator.
称为微分算子，是线性算子。

7. Evaluation Functional - 点值泛函

Let $X$ be a function space on set $M$ . The operator mapping $\phi \in X$ to $\phi(a) \in \mathbb{R}$ for $\in M$ is a linear functional called the delta-function $\delta_a$ .
设 $X$ 是集合 $M$ 上的函数空间。将 $\phi \in X$ 映射为 $\phi(a) \in \mathbb{R}$ （ $\in M$ ）的算子是线性泛函，称为点 $a$ 处的 δ 函数 $\delta_a$ 。

8. Fourier Transform Operator - 傅里叶变换算子

Let $G$ be a commutative locally compact group, $\widehat{G}$ its character group, and $L^2(G, dg)$ , $L^2(\widehat{G}, \widehat{dg})$ be Hilbert spaces. The linear operator $A$ with:
设 $G$ 是交换局部紧群， $\widehat{G}$ 为其特征标群， $L^2(G, dg)$ 和 $L^2(\widehat{G}, \widehat{dg})$ 为希尔伯特空间。线性算子 $A$ 定义为：

$\widehat{f}(\widehat{g}) = \int_G f(g) \widehat{g}(g) \, dg,$
is the Fourier transform on $L^2$ , defined everywhere in mean-square sense.
是 $L^2$ 上的傅里叶变换，在均方意义下处处定义。

Properties of Operators - 算子的性质

Inverse Operator - 逆算子

If $A$ is injective ( $\neq y \Rightarrow Ax \neq Ay$ ), the inverse operator $A^{-1}$ satisfies $A^{-1}(Ax) = x$ for $\in \operatorname{Dom}(A)$ . Existence of $A^{-1}$ relates to uniqueness of solutions to $A x = y$ .
若 $A$ 是单射（ $\neq y \Rightarrow Ax \neq Ay$ ），则逆算子 $A^{-1}$ 满足对 $\in \operatorname{Dom}(A)$ ，有 $A^{-1}(Ax) = x$ 。 $A^{-1}$ 的存在性与 $A x = y$ 解的唯一性相关。

Sum and Scalar Multiplication - 和与数乘

Sum: For operators $A, B$ with $D (A), D (B)$ , define:
和：对定义域为 $D (A), D (B)$ 的算子 $A, B$ ，定义：
$\cap D(B), \quad (A + B)x = Ax + Bx.$
Scalar Multiplication: For scalar $\lambda$ , define:
数乘：对标量 $\lambda$ ，定义：
$D(\lambda A) = D(A), \quad (\lambda A)x = \lambda(Ax).$

Operator Product - 算子乘积

The product $B A$ of operators $\to Y$ and $\to Z$ is defined by:
算子 $\to Y$ 和 $\to Z$ 的乘积 $B A$ 定义为：
$\{ x \in X : x \in D(A) \text{ and } Ax \in D(B) \}, \quad (BA)x = B(Ax).$

Projection and Involution Operators - 投影与对合算子

Projection: An operator $P$ with $P^2 = P$ is a projection operator.
投影：满足 $P^2 = P$ 的算子 $P$ 称为投影算子。
Involution: An operator $I$ with $I^2 = \mathop{\rm id}\nolimits_X$ is an involution operator.
对合：满足 $I^2 = \mathop{\rm id}\nolimits_X$ 的算子 $I$ 称为对合算子。

Applications - 应用

The theory of operators is central to functional analysis and serves as a fundamental tool in:
算子理论是泛函分析的核心，是以下领域的基本工具：

Dynamical systems;
动力系统；
Group and algebra representations;
群与代数表示；
Mathematical physics and quantum mechanics, where operators represent physical observables.
数学物理与量子力学（算子表示物理可观测量）。

Operator (Mathematics) - 算子（数学）

Short Description - 简短描述

An operator is a function acting on elements of a space (often function spaces) to produce elements of another space. It generalizes the concept of functions, often with domain/range being vector spaces or topological spaces.
算子是作用于空间（常为函数空间）元素并映射到另一空间的函数，是函数概念的推广，定义域/值域常为向量空间或拓扑空间。

Linear Operators - 线性算子

A mapping $\to V$ between vector spaces is linear if:
在向量空间间的映射 $\to V$ 是线性的，若：
$A(\alpha x + \beta y) = \alpha Ax + \beta Ay \quad \forall \alpha, \beta \in \mathbb{K}, \ x, y \in U.$
Finite-dimensional linear operators correspond to matrices under basis selection.
有限维线性算子在基下对应矩阵。

Bounded Operators - 有界算子

A linear operator $\to Y$ between normed spaces is bounded if $\exists M > 0$ such that:
赋范空间间的线性算子 $\to Y$ 是有界的，若 $\exists M > 0$ 使得：
$\|Ax\|_Y \leq M\|x\|_X \quad \forall x \in X.$
Bounded operators form a Banach algebra under the operator norm $A\| = \sup_{\|x\|=1} \|Ax\|$ .
有界算子在算子范数 $A\| = \sup_{\|x\|=1} \|Ax\|$ 下构成巴拿赫代数。

Key Examples in Analysis - 关键示例分析

Differential Operator: $\frac{d}{dx}$ on differentiable functions (unbounded).
微分算子：可微函数上的 $\frac{d}{dx}$ （无界）。
Integral Operator: $\int_0^x f(t) \, dt$ (Volterra operator, linear and bounded).
积分算子： $\int_0^x f(t) \, dt$ （沃尔泰拉算子，线性有界）。
Gradient, Divergence, Curl: Fundamental in vector calculus:
梯度、散度、旋度：矢量微积分的基本算子：

$\nabla f$ (gradient of scalar field),
$\nabla f$ （标量场的梯度），
$\nabla \cdot \mathbf{F}$ (divergence of vector field),
$\nabla \cdot \mathbf{F}$ （向量场的散度），
$\nabla \times \mathbf{F}$ (curl of vector field).
$\nabla \times \mathbf{F}$ （向量场的旋度）。

Operator vs. Function - 算子与函数的区别

Every operator is a function, but “operator” typically implies:
所有算子都是函数，但“算子”通常意味着：
Domain/range are structured spaces (e.g., vector spaces, function spaces);
定义域/值域为结构化空间（如向量空间、函数空间）；
Often linearity or continuity properties.
常具有线性或连续性等性质。
A function is a general mapping between sets, while an operator is a function with additional structural constraints on its domain and codomain.
函数是集合间的一般映射，而算子是对定义域和值域有额外结构约束的函数。

篇外：讨论

What is an operator in mathematics?

数学中的算子是什么？

Could someone please explain the mathematical difference between an operator (not in the programming sense) and a function? Is an operator a function?
能否请某人从数学角度解释算子（非编程意义上的）与函数之间的区别？算子是函数吗？

edited Feb 21, 2018 at 19:43
user168764
asked Jul 8, 2012 at 21:21
Nik

Every operator is a function, but not all functions are operators. Precisely what qualifies a function to be an operator varies with the context – it might be best to think of “operator” as a context-dependent shorthand for “function of the (particularly nice?) kind we’re considering in this subdiscipline”.
每个算子都是函数，但并非所有函数都是算子。严格来说，使函数有资格成为算子的条件因上下文而异——或许最好将“算子”视为“我们在该子学科中考虑的（特别好的？）一类函数”的上下文相关简称。
– hmakholm left over Monica
Commented Jul 8, 2012 at 21:26
Thanks @HenningMakholm. Could you please give an example? Is operator a function so frequently used that it is elevated to the status of an operator? Also, there are some properties of operators like associativity for which I am not aware of a counterpart in functions or relations. That makes me believe that operators might be fundamentally different from functions. Is that true?
感谢 @HenningMakholm。你能否举个例子？算子是否是因为被频繁使用才被提升为算子的地位？另外，算子有一些性质，比如结合律，我不清楚函数或关系中是否有对应的性质。这让我认为算子可能与函数有根本区别。这是真的吗？
– Nik
Commented Jul 8, 2012 at 21:28

@HenningMakholm Is not every function an unary operator? At least I see nothing prohibiting us from calling it such.
@HenningMakholm 难道每个函数不都是一元算子吗？至少我认为没有什么能阻止我们这样称呼它。
– user31373
Commented Jul 8, 2012 at 21:47

@Leonid: You can certainly choose to make the extra conditions on an operator void when you flesh out the concept for your field of choice. However, there are large areas of mathematics where an “operator” is usually understood to mean “linear transformation” (also known as “linear operator”), and under that convention not all functions are operators.
@Leonid：当你为所选领域阐述算子概念时，当然可以选择让算子的额外条件无效。然而，在数学的许多领域中，“算子”通常被理解为“线性变换”（也称为“线性算子”），在该约定下，并非所有函数都是算子。
– hmakholm left over Monica
Commented Jul 8, 2012 at 21:49

According to Wikipedia an operator is a function whose domain and co-domain are vector spaces (or more generally modules).
根据维基百科，算子是定义域和陪域均为向量空间（或更一般地，模）的函数。
– user73994
Commented Apr 23, 2013 at 16:16

Answers

Based on your comment it sounds like you’re actually asking about operations, not operators. A binary operation on a set $S$ is a special kind of function; namely, it is a function $\times S \to S$ . That is, it takes as input two elements of $S$ and returns another element of $S$ . We can denote such an operation by a symbol such as $\star b$ and then demand various additional properties of this operation, such as
根据你的评论，听起来你实际上问的是“运算”（operations）而非“算子”（operators）。集合 $S$ 上的二元运算是一种特殊的函数；即，它是一个 $\times S \to S$ 的函数。也就是说，它以 $S$ 的两个元素为输入，并返回 $S$ 的另一个元素。我们可以用诸如 $\star b$ 的符号表示这种运算，然后要求该运算具有各种额外性质，例如：

associativity: 结合律
$\star b) \star c = a \star (b \star c)$ ,
commutativity: 交换律
$\star b = b \star a$

and so forth. On the other hand, an arbitrary function $\to B$ between two sets only takes a single input and returns an output which is not necessarily of the same type, so one can’t speak of associativity or commutativity for such a thing. One might call a function $\to A$ a unary operation but one still can’t speak of associativity or commutativity for such a thing.
等等。另一方面，两个集合之间的任意函数 $\to B$ 仅接受单个输入并返回不一定同类型的输出，因此对于此类函数无法谈及结合律或交换律。人们可能会将函数 $\to A$ 称为一元运算，但对此类运算仍无法谈及结合律或交换律。
answered Jul 8, 2012 at 21:48
Qiaochu Yuan

On the other hand (noted because this seems to confuse the OP), we can also speak of associativity and commutativity in functional notation $f (f (a, b), c) = f (a, f (b, c))$ and $f (a, b) = f (b, a)$ . It is conventional to use the word “operation” about $f$ whenever we’re particularly interested in properties of this kind, but that linguistic convention does not express any fundamental difference in what the thing is.
另一方面（提到这一点是因为它似乎让提问者感到困惑），我们也可以用函数符号 $f (f (a, b), c) = f (a, f (b, c))$ 和 $f (a, b) = f (b, a)$ 来表述结合律和交换律。每当我们特别关注此类性质时，习惯上用“运算”一词来称呼 $f$ ，但这种语言约定并未表达该事物的任何根本区别。
– hmakholm left over Monica
Commented Jul 8, 2012 at 21:57

Thanks guys. I guess I now have a better understanding. So operations are also functions.
谢谢各位。我想我现在有了更好的理解。所以运算也是函数。
– Nik
Commented Jul 9, 2012 at 0:35
@Qiaochu Yuan, would you mind shortly explaining the difference between an operator and operation?
@Qiaochu Yuan，你介意简要解释一下算子（operator）和运算（operation）的区别吗？
– zpavlinovic
Commented Nov 16, 2012 at 21:24
@bellpeace: this should be asked as a separate question, except that I think it has already been asked (use Google, not the built-in search).
@bellpeace：这应该作为一个单独的问题提出，不过我认为它已经被问过了（用谷歌搜索，而不是内置搜索）。
– Qiaochu Yuan
Commented Nov 16, 2012 at 22:12
Not only are operations functions, but from what I gather, the converse is also true: all functions are operations. Every function is either a unary operation $\to T$ or a $k$ -ary operation $S_1 \times \dots \times S_k \to T$ . Would anyone disagree with this?
不仅运算都是函数，而且据我所知，反之亦然：所有函数都是运算。每个函数要么是一元运算 $\to T$ ，要么是 $k$ 元运算 $S_1 \times \dots \times S_k \to T$ 。有人会不同意这一点吗？
– EthanAlvaree
Commented Sep 1, 2014 at 6:49

It is a pity that mathematics (being the most exact of all sciences) has some inexact (non-standardized) terms when it comes to certain cases. The terms operator and function are used as synonymous in certain texts but the term operator is used in a narrower sense (as a special kind of function) in other texts. Hence, it is impossible to say one usage is correct and the other is incorrect. In my view
遗憾的是，数学（作为所有科学中最精确的学科）在某些情况下存在一些不精确（非标准化）的术语。“算子”和“函数”这两个术语在某些文本中被用作同义词，但在其他文本中“算子”一词被用于更狭窄的意义（作为一种特殊的函数）。因此，无法说一种用法是正确的而另一种是错误的。在我看来：

The reader of a text has to use a concept according to and only according to the definition given in that particular text (contextual meaning),
文本的读者必须且只能根据该特定文本中给出的定义（上下文含义）来使用一个概念，
International standards have to fix these kinds of problems in the future.
国际标准未来必须解决这类问题。

edited Feb 22, 2019 at 16:42
hertzsprung
answered Feb 16, 2016 at 15:53
Tadesse Bekeshie

Let $A$ and $B$ be any two sets. Then $\to B$ is said to be a function, if every element of $A$ is mapped to a unique element of $B$ . Here requirement for $A$ and $B$ is only arbitrary sets.
设 $A$ 和 $B$ 为任意两个集合。若 $A$ 的每个元素都被映射到 $B$ 的唯一元素，则称 $\to B$ 为函数。此处对 $A$ 和 $B$ 的要求仅为任意集合。

Let $V_1, V_2$ be any two vector spaces. A map or a function $V_1 \to V_2$ is an operator. Here minimum requirement of $V_1, V_2$ be vector spaces. i.e., some algebraic structure should be there in domain and co-domain.
设 $V_1, V_2$ 为任意两个向量空间。映射或函数 $V_1 \to V_2$ 是算子。此处对 $V_1, V_2$ 的最低要求是它们为向量空间，即定义域和陪域中应存在某种代数结构。
answered Jun 25, 2015 at 5:25
SKarantha