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Abstract We formulate general conditions which imply that $${\mathcal L}(X,Y)$$ , the space of operators from a Banach space X to a Banach space Y , has $$2^{{\mathfrak {c}}}$$ closed ideals, where $${\mathfrak {c}}$$ is the cardinality of the continuum. These results are applied to classical sequence spaces and Tsirelson-type spaces. In particular, we prove that the cardinality of the set ofclosed ideals in $${\mathcal L}\left (\ell _p\oplus \ell _q\right )$$ is exactly $$2^{{\mathfrak {c}}}$$ for all $$1<\infty $$ .
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Isomorphic spectrum and isomorphic length of a Banach space
We prove that, given any ordinal $$\delta < \omega_2$$, there exists a transfinite $$\delta$$-sequence of separable Banach spaces $$(X_\alpha)_{\alpha < \delta}$$ such that $$X_\alpha$$ embeds isomorphically into $$X_\beta$$ and contains no subspace isomorphic to $$X_\beta$$ for all $$\alpha < \beta < \delta$$. All these spaces are subspaces of the Banach space $$E_p = \bigl( \bigoplus_{n=1}^\infty \ell_p \bigr)_2$$, where $$1 \leq p < 2$$. Moreover, assuming Martin's axiom, we prove the same for all ordinals $$\delta$$ of continuum cardinality.
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- Award ID(s):
- 1700176
- PAR ID:
- 10285954
- Date Published:
- Journal Name:
- Carpathian Mathematical Publications
- Volume:
- 12
- Issue:
- 1
- ISSN:
- 2075-9827
- Page Range / eLocation ID:
- 88 to 93
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.15330/cmp.12.1.88-93
