Pelczynski decomposition method

The decomposition method of Pelczynski is a mathematical theorem, the evidence for existence of isomorphism between two Banach spaces is used. The theorem was proved in 1960 by the Polish mathematician Aleksander Pełczyński .

formulation

Let and two Banach spaces be such that is isomorphic to a complemented , closed subspace of space and is in turn isomorphic to a complemented, closed subspace of . Furthermore, one of the following conditions is met: ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Y}$ ${\displaystyle X}$

a) and , ${\displaystyle X\cong X\oplus X}$ ${\displaystyle Y\cong Y\oplus Y}$

b) for a certain or . ${\displaystyle X\cong \ell ^{p}(X)}$ ${\displaystyle p\in [1,\infty )}$ ${\displaystyle X\cong c_{0}(X)}$

Then the space is isomorphic to . ${\displaystyle X}$ ${\displaystyle Y}$

The above symbols and denote the ℓ ^p sum or c ₀ sum of countably many copies of space . ${\displaystyle \ell ^{p}(X)}$ ${\displaystyle c_{0}(X)}$ ${\displaystyle X}$

proof

Be and for certain Banach spaces and . Under the condition a) isomorphisms exist ${\displaystyle X\cong Y\oplus E}$ ${\displaystyle Y\cong X\oplus F}$ ${\displaystyle E}$ ${\displaystyle F}$

${\displaystyle X\cong Y\oplus E\cong Y\oplus Y\oplus E\cong Y\oplus X}$

and the same

${\displaystyle Y\cong X\oplus F\cong X\oplus X\oplus F\cong X\oplus Y}$,

so overall ${\displaystyle X\cong Y}$

Under the condition b) applies in particular and thus . So it applies ${\displaystyle X\cong X\oplus X}$ ${\displaystyle Y\cong X\oplus F\cong X\oplus X\oplus F\cong X\oplus Y}$

${\displaystyle X\cong \ell ^{p}(X)\cong \ell ^{p}(Y\oplus E)\cong \ell ^{p}(Y)\oplus \ell ^{p}(E)\cong Y\oplus \ell ^{p}(Y)\oplus \ell ^{p}(E)\cong Y\oplus X\cong X\oplus Y\cong Y}$.

An analogous proof results for . ${\displaystyle X\cong c_{0}(X)}$

Application examples

• Using Pelczynski's decomposition method, one can show that every infinite-dimensional, complemented sub-annach space is isomorphic from or to the original space . ${\displaystyle c_{0}}$ ${\displaystyle \ell ^{p}}$
• Using Pelczynski's decomposition method one can prove that the Banach spaces and L ^∞ ([0,1]) are isomorphic, but they are not isometrically isomorphic . ${\displaystyle \ell ^{\infty }}$

Remarks

• Timothy Gowers has shown that there are a pair of Banach spaces and such that is isomorphic to a complemented, closed subspace of and isomorphic to a complemented, closed subspace of , the spaces and are not isomorphic. Additional requirements such as a) or b) cannot be dispensed with in the above sentence. This is the negative solution to the so-called Schröder-Bernstein problem for Banach spaces. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle Y}$
• Piotr Koszmider has a pair of totally unrelated compact spaces and constructed so that isometric isomorphic to a complemented, closed subspace of , and vice versa, but the Banach and are not isomorphic. ${\displaystyle K_{1}}$ ${\displaystyle K_{2}}$ ${\displaystyle C(K_{1})}$ ${\displaystyle C(K_{2})}$ ${\displaystyle C(K_{1})}$ ${\displaystyle C(K_{2})}$
• Valentin Ferenczi and Elói Medina Galego have constructed a continuum of pairwise non-isomorphic Banach spaces, so that for each pair and from this class is isomorphic to a complemented, closed subspace of and isomorphic to a complemented, closed subspace of . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Y}$ ${\displaystyle X}$
• Further generalizations of the Pelczynski decomposition method can be found in the literature.

Individual evidence

1. ^ A. Pelczynski: Projections in certain Banach Spaces , Studia Math. (1960), Volume 19, pages 209-228.
2. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory , Springer-Verlag (2006), ISBN 978-1-4419-2099-7 , pages 34-36
3. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory , Springer-Verlag (2006), ISBN 978-1-4419-2099-7 , Theorem 2.2.4
4. ^ A. Pełczyński: On the isomorphism of the spaces m and M , Bull. Acad. Pole. Sci. (1958), Volume 6, pp. 695-696
5. ^ WT Gowers: A solution to the Schroeder-Bernstein problem for Banach spaces , Bull. London Math. Soc. (1996), Vol. 28, pp. 297-304
6. ↑ P. Koszmider: AC (K) Banach space which does not have the Schroeder-Bernstein property , Studia Math. (2012), Volume 212, Pages 95-117, arxiv : 1106.2917 .
7. ^ V. Ferenczi, EM Galego: Some results about the Schroeder-Bernstein Property for separable Banach spaces , Canad. J. Math. (2007) Volume 591, pages 63-84.
8. ^ EM Galego: Generalizations of Pełczyński's decomposition method for Banach spaces containing a complemented copy of their squares , Archiv der Mathematik (2008), Vol. 90-6, pp. 530-536. doi: 10.1007 / s00013-008-2568-1
9. ^ EM Galego: Towards a maximal extension of Pełczyński's decomposition method in Banach spaces , Journal of Mathematical Analysis and Applications (2009), Volume 356-1, pp. 86-95.