Theorem of Kato

The set of Kato is a mathematical theorem which the area of the functional analysis belongs and on the Japanese mathematician Tosio Kato back. The theorem deals with a property of continuous linear mappings between Banach spaces .

Formulation of the sentence

The sentence can be stated as follows:

Let two Banach spaces and a continuous linear mapping be given . ${\displaystyle E,F}$ ${\displaystyle A\colon E\to F}$

The image space has finite codimension . ${\displaystyle A(E)\subseteq F}$

Then:

${\displaystyle A(E)}$is a closed subspace of . ${\displaystyle F}$

generalization

Kato's theorem is a direct consequence of a more general theorem, which reads as follows:

If, under the above conditions, there is a closed subspace such that on the one hand and on the other hand the direct sum is a closed subspace of , then it must itself be a closed subspace of . ${\displaystyle G\subseteq F}$ ${\displaystyle A(E)\cap G=\{0\}}$ ${\displaystyle A(E)\oplus G}$ ${\displaystyle F}$ ${\displaystyle A(E)}$ ${\displaystyle F}$

Different version

Kato's theorem can also be found in the specialist literature in another version, which is located between the version presented above and the above generalization. This version reads as follows:

Let it be a continuous linear endomorphism on the Banach space and further a closed subspace of such that on the one hand and on the other hand the direct sum is a closed subspace of . ${\displaystyle A}$ ${\displaystyle E}$ ${\displaystyle G}$ ${\displaystyle E}$ ${\displaystyle A(E)\cap G=\{0\}}$ ${\displaystyle A(E)\oplus G}$ ${\displaystyle E}$

Then the image space is itself a closed subspace of . ${\displaystyle A(E)}$ ${\displaystyle E}$

Related result: Riesz's theorem on compact operators

The importance of the question raised in Kato's theorem about the connection between isolation and codimensionality of the image spaces of continuous linear mappings is also evident when examining the compact operators on Banach spaces. A classic sentence by the Hungarian mathematician F. Riesz applies to this :

Let it be a compact operator on the Banach space . ${\displaystyle A}$ ${\displaystyle E}$

Then the associated operator has the following properties: ${\displaystyle T=Id-A}$

(1) The null space of is finite - dimensional . ${\displaystyle T}$

(2) The image space of is closed. ${\displaystyle T}$

(3) The factor space is finite-dimensional. ${\displaystyle E/T(E)}$

Remarks

1. Harro Heuser calls the generalization of Kato's theorem an important sufficient criterion .
2. In the other version above, Kato's theorem plays an important role in spectral theory, for example .
3. In English-speaking countries, the Kato theorem is sometimes referred to as the closed range theorem of T. Kato .

literature

• Harro Heuser: Functional Analysis. Theory and application (=  mathematical guidelines ). 4th revised edition. BG Teubner Verlag, Wiesbaden 2006, ISBN 978-3-8351-0026-8 ( MR2380292 ).
• Friedrich Hirzebruch , Winfried Scharlau : Introduction to functional analysis (=  BI university pocket books . Volume 296 ). Bibliographical Institute, Mannheim / Vienna / Zurich 1971, ISBN 3-411-00296-4 ( MR0463864 ).
• Hans-Dieter Wacker: About the generalization of a sentence by Kato . In: Mathematical Journal . tape 190 , 1985, pp. 55-61 , doi : 10.1007 / BF01159163 ( MR0793348 ).

Individual evidence

1. ↑ Harro Heuser: functional analysis. 2006, pp. 309-310
2. ↑ ^{a b c} Harro Heuser: functional analysis. 2006, p. 310
3. ↑ ^{a b} Hans-Dieter Wacker: About the generalization of a theorem by Kato. In: Mathematische Zeitschrift 190 (1985), pp. 55 ff
4. ↑ Friedrich Hirzebruch, Winfried Scharlau: Introduction to Functional Analysis. 1971, p. 103 ff
5. ↑ Cf. MR0793348  ( page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice. !