合成算子 (Koopman算子)

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有关组合运算符 ∘ 的信息，请参阅复合函数和关系组合。

In mathematics, the composition operator [math]\displaystyle{ C_\phi }[/math] with symbol [math]\displaystyle{ \phi }[/math] is a linear operator defined by the rule [math]\displaystyle{ C_\phi (f) = f \circ \phi }[/math] where [math]\displaystyle{ f \circ \phi }[/math] denotes function composition.

在数学mathematics中，带有符号 [math]\displaystyle{ \phi }[/math] 的复合算子composition operator[math]\displaystyle{ C_\phi }[/math]是一个线性算子linear operator，定义为 [math]\displaystyle{ C_\phi (f) = f \circ \phi }[/math]，其中[math]\displaystyle{ f \circ \phi }[/math]表示复合函数function composition。

The study of composition operators is covered by AMS category 47B33.

复合算子研究属于 AMS 第47b33类。

在物理学中In physics

In physics, and especially the area of dynamical systems, the composition operator is usually referred to as the Koopman operator^[1]^[2] (and its wild surge in popularity^[3] is sometimes jokingly called "Koopmania"^[4]), named after Bernard Koopman. It is the left-adjoint of the transfer operator of Frobenius–Perron.

在物理学physics中，特别是在动力系统dynamical systems领域，组合运算符通常被称为库普曼运算符Koopman operator^[1]^[2](它的疯狂流行^[3]有时被戏称为"库普曼狂热 ”^[4])，以伯纳德·库普曼Bernard Koopman命名。它是Frobenius-Perron的传输算子transfer的左伴随left-adjoint。

在 Borel函数演算中In Borel functional calculus

Using the language of category theory, the composition operator is a pull-back on the space of measurable functions; it is adjoint to the transfer operator in the same way that the pull-back is adjoint to the push-forward; the composition operator is the inverse image functor.

利用范畴论category theory的语言，复合算子是对可测函数measurable functions空间的拉回pull-back；它与传输算子transfer operator相伴随，就像拉回与推进push-forward相伴随一样；复合算子是逆像函子inverse image functor。

Since the domain considered here is that of Borel functions, the above describes the Koopman operator as it appears in Borel functional calculus.

由于这里考虑的是Borel函数Borel functions的定义域，所以上面描述的 Koopman 算子出现在 Borel 函数演算Borel functional calculus中。

在全纯函数演算中In holomorphic functional calculus

The domain of a composition operator can be taken more narrowly, as some Banach space, often consisting of holomorphic functions: for example, some Hardy space or Bergman space. In this case, the composition operator lies in the realm of some functional calculus, such as the holomorphic functional calculus.

一个复合算子的定义域domain可以取得更窄一些，如某些 Banach 空间Banach space，通常由全纯函数holomorphic functions组成: 例如某些 Hardy 空间Hardy space或 Bergman 空间Bergman space。在这种情况下，复合算子处于一些函数演算functional calculus的领域，例如全纯函数演算holomorphic functional calculus。

Interesting questions posed in the study of composition operators often relate to how the spectral properties of the operator depend on the function space. Other questions include whether [math]\displaystyle{ C_\phi }[/math] is compact or trace-class; answers typically depend on how the function [math]\displaystyle{ \varphi }[/math] behaves on the boundary of some domain.

在复合算子的研究中，有趣的问题往往与算子的谱性质spectral properties如何依赖于函数空间function space有关。其他问题包括 [math]\displaystyle{ C_\phi }[/math]是紧空间compact的还是迹类算子trace-class; 答案通常取决于函数[math]\displaystyle{ \varphi }[/math]在某个域边界boundary上的行为。

When the transfer operator is a left-shift operator, the Koopman operator, as its adjoint, can be taken to be the right-shift operator. An appropriate basis, explicitly manifesting the shift, can often be found in the orthogonal polynomials. When these are orthogonal on the real number line, the shift is given by the Jacobi operator.^[5] When the polynomials are orthogonal on some region of the complex plane (viz, in Bergman space), the Jacobi operator is replaced by a Hessenberg operator.^[6]

当传输算子是左移算子left-shift operator时，作为它的伴随算子的库普曼算子可以被认为是右移算子。在正交多项式orthogonal polynomials中，常常可以找到一组适当的基，它能显性地表示这种位移。当它们在实数线上正交时，位移由雅可比算子Jacobi operator^[5]给出。当多项式在复平面的某个区域上正交时(即在伯格曼空间Bergman space中)，雅可比算子被海森伯格算子Hessenberg operator所取代。^[6]

应用Applications

In mathematics, composition operators commonly occur in the study of shift operators, for example, in the Beurling–Lax theorem and the Wold decomposition. Shift operators can be studied as one-dimensional spin lattices. Composition operators appear in the theory of Aleksandrov–Clark measures.

在数学中，复合算子常出现在移位算子shift perator的研究中，如Beurling-Lax定理Beurling-Lax theorm和Wold分解Wold decomposition。移位算符可以作为一维自旋晶格spin lattice来研究。复合算子出现在Aleksandrov-Clark测度理论Aleksandrov-Clark measure中。

The eigenvalue equation of the composition operator is Schröder's equation, and the principal eigenfunction [math]\displaystyle{ f(x) }[/math] is often called Schröder's function or Koenigs function.

复合算子的特征值eigenvalue方程是施罗德的方程Schröder's equation，主特征函数eigenfunction[math]\displaystyle{ f(x) }[/math]通常被称为施罗德函数Schröder function或柯尼希函数Koenigs function。

参见See also

模板:Annotated link

References

↑ ^1.0 ^1.1 Koopman, B. O. (1931). "Hamiltonian Systems and Transformation in Hilbert Space". Proceedings of the National Academy of Sciences. 17 (5): 315–318. Bibcode:1931PNAS...17..315K. doi:10.1073/pnas.17.5.315. PMC 1076052. PMID 16577368.
↑ ^2.0 ^2.1 Gaspard, Pierre (1998). Chaos, scattering and statistical mechanics. Cambridge University Press. doi:10.1017/CBO9780511628856. ISBN 978-0-511-62885-6.
↑ ^3.0 ^3.1 Budišić, Marko, Ryan Mohr, and Igor Mezić. "Applied koopmanism." Chaos: An Interdisciplinary Journal of Nonlinear Science 22, no. 4 (2012): 047510. https://doi.org/10.1063/1.4772195
↑ ^4.0 ^4.1 Shervin Predrag Cvitanović, Roberto Artuso, Ronnie Mainieri, Gregor Tanner, Gábor Vattay, Niall Whelan and Andreas Wirzba, Chaos: Classical and Quantum Appendix H version 15.9, (2017), http://chaosbook.org/version15/chapters/appendMeasure.pdf

↑ ^5.0 ^5.1 Gerald Teschl, "Jacobi Operators and Completely Integrable Nonlinear Lattices" (2000) American Mathematical Society. https://www.mat.univie.ac.at/~gerald/ftp/book-jac/jacop.pdf

↑ ^6.0 ^6.1 Tomeo, V.; Torrano, E. (2011). "Two applications of the subnormality of the Hessenberg matrix related to general orthogonal polynomials". Linear Algebra and Its Applications. 435 (9): 2314–2320. doi:10.1016/j.laa.2011.04.027.

C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, Florida, 1995. xii+388 pp.

J. H. Shapiro, Composition operators and classical function theory. Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993. xvi+223 pp.

.

C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, Florida, 1995. xii+388 pp. .
J. H. Shapiro, Composition operators and classical function theory. Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993. xvi+223 pp. .

c. c. Cowen 和 b. d. MacCluer，解析函数空间上的复合算子。高等数学研究。1995. xii + 388 pp..
j · h · 夏皮罗，复合算子与经典函数论。大学: 数学教材。1993年，纽约，Springer-Verlag，xvi + 223页。.

模板:Functional Analysis 模板:TopologicalVectorSpaces

Category:Functional analysis Category:Dynamical systems Category:Operator theory Category:Topological vector spaces

范畴: 函数分析范畴: 动力系统范畴: 算子理论范畴: 拓扑向量空间

This page was moved from wikipedia:en:Composition operator. Its edit history can be viewed at 合成算子 (Koopman算子)/edithistory

References

C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, Florida, 1995. xii+388 pp.

.

J. H. Shapiro, Composition operators and classical function theory. Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993. xvi+223 pp.

.

模板:Functional Analysis 模板:TopologicalVectorSpaces

[:0-1] 1.0 ^1.1 Koopman, B. O. (1931). "Hamiltonian Systems and Transformation in Hilbert Space". Proceedings of the National Academy of Sciences. 17 (5): 315–318. Bibcode:1931PNAS...17..315K. doi:10.1073/pnas.17.5.315. PMC 1076052. PMID 16577368.

[:1-2] 2.0 ^2.1 Gaspard, Pierre (1998). Chaos, scattering and statistical mechanics. Cambridge University Press. doi:10.1017/CBO9780511628856. ISBN 978-0-511-62885-6.

[:2-3] 3.0 ^3.1 Budišić, Marko, Ryan Mohr, and Igor Mezić. "Applied koopmanism." Chaos: An Interdisciplinary Journal of Nonlinear Science 22, no. 4 (2012): 047510. https://doi.org/10.1063/1.4772195

[:3-4] 4.0 ^4.1 Shervin Predrag Cvitanović, Roberto Artuso, Ronnie Mainieri, Gregor Tanner, Gábor Vattay, Niall Whelan and Andreas Wirzba, Chaos: Classical and Quantum Appendix H version 15.9, (2017), http://chaosbook.org/version15/chapters/appendMeasure.pdf

[:4-5] 5.0 ^5.1 Gerald Teschl, "Jacobi Operators and Completely Integrable Nonlinear Lattices" (2000) American Mathematical Society. https://www.mat.univie.ac.at/~gerald/ftp/book-jac/jacop.pdf
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[:5-6] 6.0 ^6.1 Tomeo, V.; Torrano, E. (2011). "Two applications of the subnormality of the Hessenberg matrix related to general orthogonal polynomials". Linear Algebra and Its Applications. 435 (9): 2314–2320. doi:10.1016/j.laa.2011.04.027.

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合成算子 (Koopman算子)

目录

在物理学中In physics

在 Borel函数演算中In Borel functional calculus

在全纯函数演算中In holomorphic functional calculus

应用Applications

参见See also

References

References

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