Interpolative construction for operator ideals
Abstract
The problem from which this article originated is the following: given an operator
between Banach spaces belonging simultaneously to two operator ideals,
and
say, when is it possible to find a decomposition
, where
and
, or at least
and
, with
and
being associated with
and
in a specific sense? It was shown by S. Heinrich [2] that such a decomposition is always possible, with
and
,if
and
are uniformly closed,
is surjective, and
is injective.Heinrich’s arguments are based on a simple interpolation technique which appears to be strongy related to certain general constructions with operator ideals that were successfully applied in a seemingly different context in recent years (ref.[8],[5],and [4]-[7], [1]). We intend to investigate the fundamentals of such constructions and their interpolation-theoretic background in this paper, with emphasis on the impact to the factorization problem.Applications will be given for ideals generated by s-number sequences and to type p and cotype q operators.
![T:E→ F](http://siba-ese.unile.it/plugins/generic/latexRender/cache/51e9a816b01fdae9d76d4d58b3fdfddf.png)
![\mathcal A](http://siba-ese.unile.it/plugins/generic/latexRender/cache/861df74596abb976c25bcec0d09e08c9.png)
![\mathcal B](http://siba-ese.unile.it/plugins/generic/latexRender/cache/ca2131fc805663dd83f22eaaaf58ad99.png)
![T = A· B](http://siba-ese.unile.it/plugins/generic/latexRender/cache/153793472c1f269a9a629bc08d7d08c9.png)
![A∈ \mathcal A](http://siba-ese.unile.it/plugins/generic/latexRender/cache/9a1152ff5d742c67d030aaf86a90ee46.png)
![B∈\mathcal B](http://siba-ese.unile.it/plugins/generic/latexRender/cache/e2050c96218e637c719869f84dfe6ddc.png)
![A∈ \dot{\mathcal{A}}](http://siba-ese.unile.it/plugins/generic/latexRender/cache/9c5934ec3b3510fd45cc33c648033ac1.png)
![B∈ \ddot{\mathcal B}](http://siba-ese.unile.it/plugins/generic/latexRender/cache/630e4752457c4e3cb11080b23212aca4.png)
![\dot{\mathcal{A}}](http://siba-ese.unile.it/plugins/generic/latexRender/cache/79e97cfbc133a4bde696124a91a49510.png)
![\ddot{\mathcal B}](http://siba-ese.unile.it/plugins/generic/latexRender/cache/4de0a321f22180d98cba9b2b7bd44fcf.png)
![\mathcal A](http://siba-ese.unile.it/plugins/generic/latexRender/cache/861df74596abb976c25bcec0d09e08c9.png)
![\mathcal B](http://siba-ese.unile.it/plugins/generic/latexRender/cache/ca2131fc805663dd83f22eaaaf58ad99.png)
![\mathcal A=\dot{\mathcal{A}}](http://siba-ese.unile.it/plugins/generic/latexRender/cache/9761cdc7bc051cb0d0badb63cdc01741.png)
![\mathcal B=\ddot{\mathcal B}](http://siba-ese.unile.it/plugins/generic/latexRender/cache/ebf7632840b90c810f337b4f3d2ac3bc.png)
![\mathcal A](http://siba-ese.unile.it/plugins/generic/latexRender/cache/861df74596abb976c25bcec0d09e08c9.png)
![\mathcal B](http://siba-ese.unile.it/plugins/generic/latexRender/cache/ca2131fc805663dd83f22eaaaf58ad99.png)
![\mathcal A](http://siba-ese.unile.it/plugins/generic/latexRender/cache/861df74596abb976c25bcec0d09e08c9.png)
![\mathcal B](http://siba-ese.unile.it/plugins/generic/latexRender/cache/ca2131fc805663dd83f22eaaaf58ad99.png)
DOI Code:
10.1285/i15900932v8n1p45
Full Text: PDF