# 无界算子

• “无界”有时可以被理解为 "无需有界"，或者說 "不一定有界"；
• “算符”当被理解为“线性算符”（这和“有界算子”是相同的）；
• 算符的定义域为线性子空间, 不必为全空间；
• 线性子空间不必有界； 一般被假定为稠密
• 特殊情况下的有界算子，定义域被假定为全空间

## 定义与基本性质

B1B2巴拿赫空间. 无界算子 (或简称为算子) T : B1B2是一个线性映射 T， 从B1 的线性子空间D(T) （T的定义域）映射到空间 B2.[5] 不同于惯例, T 可能不定义在整个空间B1.

${\displaystyle \|x\|_{T}={\sqrt {\|x\|^{2}+\|Tx\|^{2}}}\ .}$

T : B1B2为闭集, 在它的定义域上稠密且连续, 则它定义在B1上.[8]

## 参考资料

• Berezansky, Y.M.; Sheftel, Z.G.; Us, G.F., Functional analysis II, Birkhäuser, 1996 (see Chapter 12 "General theory of unbounded operators in Hilbert spaces").
• Brezis, Haïm, Analyse fonctionnelle — Théorie et applications, Paris: Mason, 1983 （法语）
• Hazewinkel, Michiel (编), Unbounded operator, 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4
• Hall, B.C., Chapter 9. Unbounded Self-adjoint Operators, Quantum Theory for Mathematicians, Graduate Texts in Mathematics, Springer, 2013
• Kato, Tosio, Chapter 5. Operators in Hilbert Space, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, 1995, ISBN 3-540-58661-X
• Pedersen, Gert K., Analysis now, Springer, 1989 (see Chapter 5 "Unbounded operators").
• Reed, Michael; Simon, Barry, Methods of Modern Mathematical Physics, 1: Functional Analysis revised and enlarged, Academic Press, 1980 (see Chapter 8 "Unbounded operators").
• Yoshida, Kôsaku, Functional Analysis sixth, Springer, 1980

1. ^ Reed & Simon 1980，Notes to Chapter VIII, page 305
2. ^ von Neumann, J., Allgemeine Eigenwerttheorie Hermitescher Functionaloperatoren (General Eigenvalue Theory of Hermitian Functional Operators), Mathematische Annalen, 1930, 102 (1): 49–131, doi:10.1007/BF01782338
3. ^ Stone, Marshall Harvey. Linear Transformations in Hilbert Space and Their Applications to Analysis. Reprint of the 1932 Ed. American Mathematical Society. 1932 [2014-03-29]. ISBN 978-0-8218-7452-3. （原始内容存档于2014-06-29）.
4. ^ von Neumann, J., Über Adjungierte Funktionaloperatore (On Adjoint Functional Operators), Annals of Mathematics, Second Series, 1936, 33 (2): 294–310, JSTOR 1968331, doi:10.2307/1968331
5. ^ Pedersen 1989，5.1.1
6. Pedersen 1989，5.1.4
7. ^
8. ^ Suppose fj is a sequence in the domain of T that converges to gB1. Since T is uniformly continuous on its domain, Tfj is Cauchy in B2. Thus, (fj, Tfj) is Cauchy and so converges to some (f, Tf) since the graph of T is closed. Hence, f = g, and the domain of T is closed.
9. 引用错误：没有为名为Pedersen-5.1.12的参考文献提供内容