Abstract
A bounded linear operator acting between Banach spaces is called a Fredholm operator if the dimension of its kernel and the codimension of its image are both finite. An equation defined by a Fredholm operator sometimes enjoys a nice property which reduces the difficulties associated with infinite dimension to the finite dimensional problem. The object of this chapter is to study the basic elements of Fredholm operators, which will be made use of in the next chapter in the context of bifurcation theory.
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Notes
- 1.
- 2.
Let \(P:\mathfrak {M}\rightarrow \mathfrak {M}\) the identity mapping on \(\mathfrak {M}\). This P is clearly contained in \(\mathbb {P}\).
- 3.
- 4.
\(\mathfrak {X}'\) is the dual space of \(\mathfrak {X}\). 〈Λ i, x j〉 means the value of Λ j at x j; i.e. Λ j(x j). δ ij is Kronecker’s delta.
- 5.
- 6.
\(\mathfrak {M}^\bot \) denotes the annihilator, i.e.
$$\displaystyle \begin{aligned} \mathfrak{M}^\perp=\{\varLambda\in \mathfrak{X}' | \varLambda x=0 \quad \text{for all}\; x\in \mathfrak{M}\}. \end{aligned}$$ - 7.
Set α 1 Λ 1 + ⋯ + α n Λ n = 0. Evaluate the left-hand side at v j. By the definition of a biorthogonal system, it follows that α j = 0.
- 8.
The topologies of \(\mathfrak {C}^1\) and \(\mathfrak {C}\) are defined by the \(\mathfrak {C}^1\)-norm and the uniform convergence norm, respectively.
- 9.
Let \(A: \mathfrak {X}\rightarrow \mathfrak {X}\) be a compact operator and λ ≠ 0. All the following four numbers are finite and equal. (A ′ is the dual operator of A.)
$$\displaystyle \begin{aligned} \alpha &=\dim \mathrm{Ker}(A-\lambda I), \quad \beta=\mathrm{codim}(A-\lambda I)(\mathfrak{X}), \\ \alpha^{\prime}&=\dim \mathrm{Ker}(A^{\prime}-\lambda L), \quad \beta^{\prime}=\mathrm{codim}(A^{\prime}-\lambda I)(\mathfrak{X}').\end{aligned} $$ - 10.
We can not have recourse to Theorem 5.1 because the closedness of \(T(\mathfrak {X})\) has not been shown yet.
- 11.
- 12.
K′ is the dual operator of K.
- 13.
The reason is as follows. For any ε > 0, there exists some \(v_\varepsilon \in \mathfrak {M}\) such that ∥v ε∥ > ε −1∥(I + K)v ε∥. So defining m ε = v ε∕∥v ε∥, we have
$$\displaystyle \begin{aligned} \|(I+K)m_\varepsilon \|<\varepsilon . \end{aligned}$$ - 14.
See footnote 11, p. 13.
- 15.
In the case of ∥S∥ = 0, we may consider ε = ∞. But such a case occurs when \(\dim \mathfrak {X}<\infty \) and \(\dim \mathfrak {Y}<\infty \).
- 16.
Maruyama [4] pp. 153–154.
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Kato, T.: Iso Kaiseki (Functional Analysis). Kyoritsu Shuppan, Tokyo (1957) (Originally published in Japanese)
Kuroda, N.: Kansu Kaiseki (Functional Analysis). Kyoritsu Shuppan, Tokyo (1980) (Originally published in Japanese)
Maruyama, T.: Suri-keizaigaku no Hoho (Methods in Mathematical Economics). Sobunsha, Tokyo (1995) (Originally published in Japanese)
Yosida, K.: Functional Analysis. 3rd edn. Springer, New York (1971)
Zeidler, E.: Applied Functional Analysis. Springer, New York (1995)
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Maruyama, T. (2018). Fredholm Operators. In: Fourier Analysis of Economic Phenomena. Monographs in Mathematical Economics, vol 2. Springer, Singapore. https://doi.org/10.1007/978-981-13-2730-8_10
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