In functional analysis, compact operators are linear operators on Banach spaces that map bounded sets to relatively compact sets. In the case of a Hilbert space H, the compact operators are the closure of the finite rank operators in the uniform operator topology. In general, operators on infinite-dimensional spaces feature properties that do not appear in the finite-dimensional case, i.e. for matrices. The compact operators are notable in that they share as much similarity with matrices as one can expect from a general operator. In particular, the spectral properties of compact operators resemble those of square matrices.
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| - コンパクト作用素のスペクトル理論 (ja)
- Spectral theory of compact operators (en)
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| - 数学の函数解析学の分野におけるコンパクト作用素のスペクトル理論（コンパクトさようそのスペクトルりろん、英: spectral theory of compact operators）は、リース・フリジェシュによって初めて構築された。コンパクト作用素は有界集合を相対コンパクト集合に写すバナッハ空間上の線型作用素である。ヒルベルト空間 H の場合、コンパクト作用素は一様作用素位相における有限ランクの作用素の閉包である。一般に無限次元空間上の作用素は、有限次元の場合、すなわち行列の場合では現れない性質を示す。コンパクト作用素は、一般の作用素から期待される以上に行列との共通点を多く持つという点において価値がある。特に、コンパクト作用素のスペクトル性は正方行列のそれと似ている。 この記事では、初めに行列の場合の対応する結果をまとめた後、コンパクト作用素のスペクトル性について議論する。読者は、殆どの内容が一つ一つ行列の場合に対応することに気付くであろう。 (ja)
- In functional analysis, compact operators are linear operators on Banach spaces that map bounded sets to relatively compact sets. In the case of a Hilbert space H, the compact operators are the closure of the finite rank operators in the uniform operator topology. In general, operators on infinite-dimensional spaces feature properties that do not appear in the finite-dimensional case, i.e. for matrices. The compact operators are notable in that they share as much similarity with matrices as one can expect from a general operator. In particular, the spectral properties of compact operators resemble those of square matrices. (en)
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| - Theorem (en)
- Lemma 1 (en)
- Lemma 2 (en)
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| proof
| - Let xn → y in norm.
If {xn} is bounded, then compactness of C implies that there exists a subsequence xnk such that C xnk is norm convergent.
So xnk = xnk + C xnk is norm convergent, to some x.
This gives xnk → x = y.
The same argument goes through if the distances d is bounded.
But d must be bounded. Suppose this is not the case.
Pass now to the quotient map of , still denoted by , on X/Ker.
The quotient norm on X/Ker is still denoted by ·, and {xn} are now viewed as representatives of their equivalence classes in the quotient space.
Take a subsequence {xnk} such that xnk > k and define a sequence of unit vectors by znk = xnk/xnk.
Again we would have znk → z for some z.
Since znk = xnk/ xnk → 0, we have z = 0 i.e. z ∈ Ker.
Since we passed to the quotient map, z = 0.
This is impossible because z is the norm limit of a sequence of unit vectors.
Thus the lemma is proven. (en)
- i) Without loss of generality, assume λ = 1. λ ∈ σ not being an eigenvalue means is injective but not surjective.
By Lemma 2, Y1 = Ran is a closed proper subspace of X.
Since is injective, Y2 = Y1 is again a closed proper subspace of Y1.
Define Yn = Ran'n. Consider the decreasing sequence of subspaces
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where all inclusions are proper. By lemma 1, we can choose unit vectors yn ∈ Yn such that d > ½.
Compactness of C means {C yn} must contain a norm convergent subsequence. But for n < m
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and notice that
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which implies Cyn − Cym > ½.
This is a contradiction, and so λ must be an eigenvalue.
ii) The sequence { Yn = Ker'n} is an increasing sequence of closed subspaces.
The theorem claims it stops. Suppose it does not stop, i.e. the inclusion Ker'n ⊂ Ker'n+1 is proper for all n.
By lemma 1, there exists a sequence {yn}n ≥ 2 of unit vectors such that yn ∈ Y'n and d > ½.
As before, compactness of C means {C yn} must contain a norm convergent subsequence. But for n < m
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and notice that
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which implies Cyn − Cym > ½.
This is a contradiction, and so the sequence { Yn = Ker'n} must terminate at some finite m.
Using the definition of the Kernel, we can show that the unit sphere of Ker is compact, so that Ker is finite-dimensional.
Ker'n is finite-dimensional for the same reason.
iii) Suppose there exist infinite distinct eigenvalues {λn}, with corresponding eigenvectors {xn}, such that λn > ε for all n.
Define Yn = span{x1...xn}.
The sequence {Yn} is a strictly increasing sequence.
Choose unit vectors such that yn ∈ Y'n and d > ½. Then for n < m
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But
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therefore Cyn − Cym > ε/2, a contradiction.
So we have that there are only finite distinct eigenvalues outside any ball centered at zero.
This immediately gives us that zero is the only possible limit point of eigenvalues and there are at most countable distinct eigenvalues .
iv) This is an immediate consequence of iii). The set of eigenvalues {λ} is the union
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Because σ is a bounded set and the eigenvalues can only accumulate at 0, each Sn is finite, which gives the desired result.
v) As in the matrix case, this is a direct application of the holomorphic functional calculus. (en)
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| - In functional analysis, compact operators are linear operators on Banach spaces that map bounded sets to relatively compact sets. In the case of a Hilbert space H, the compact operators are the closure of the finite rank operators in the uniform operator topology. In general, operators on infinite-dimensional spaces feature properties that do not appear in the finite-dimensional case, i.e. for matrices. The compact operators are notable in that they share as much similarity with matrices as one can expect from a general operator. In particular, the spectral properties of compact operators resemble those of square matrices. This article first summarizes the corresponding results from the matrix case before discussing the spectral properties of compact operators. The reader will see that most statements transfer verbatim from the matrix case. The spectral theory of compact operators was first developed by F. Riesz. (en)
- 数学の函数解析学の分野におけるコンパクト作用素のスペクトル理論（コンパクトさようそのスペクトルりろん、英: spectral theory of compact operators）は、リース・フリジェシュによって初めて構築された。コンパクト作用素は有界集合を相対コンパクト集合に写すバナッハ空間上の線型作用素である。ヒルベルト空間 H の場合、コンパクト作用素は一様作用素位相における有限ランクの作用素の閉包である。一般に無限次元空間上の作用素は、有限次元の場合、すなわち行列の場合では現れない性質を示す。コンパクト作用素は、一般の作用素から期待される以上に行列との共通点を多く持つという点において価値がある。特に、コンパクト作用素のスペクトル性は正方行列のそれと似ている。 この記事では、初めに行列の場合の対応する結果をまとめた後、コンパクト作用素のスペクトル性について議論する。読者は、殆どの内容が一つ一つ行列の場合に対応することに気付くであろう。 (ja)
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| math statement
| - Let X be a Banach space, C be a compact operator acting on X, and σ be the spectrum of C.
Every nonzero λ ∈ σ is an eigenvalue of C.
For all nonzero λ ∈ σ, there exist m such that Ker = Ker, and this subspace is finite-dimensional.
The eigenvalues can only accumulate at 0. If the dimension of X is not finite, then σ must contain 0.
σ is at most countably infinite.
Every nonzero λ ∈ σ is a pole of the resolvent function ζ → −1. (en)
- If C is compact, then Ran is closed. (en)
- Let X be a Banach space and Y ⊂ X, Y ≠ X, be a closed subspace.
For all ε > 0, there exists x ∈ X such that x = 1 and
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where d is the distance from x to Y. (en)
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