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Quasi-normed operator ideals on Banach n-tuples. (English) Zbl 0655.47040
Let $$<E_ 1,...,E_ n>$$ be a Banach n-tuple, i.e. $$E_ i$$ is a Banach space continuously embedded in some topological Hausdorff space, $$i=1,...,n$$. In complete analogy to the well-known theory of operator ideals on Banach spaces [see e.g. A. Pietsch, Operator ideals (1980; Zbl 0399.47039)] the authors define operator ideals $${\mathcal A}$$ on Banach n-tuples and quasi-normed operator ideals $$<{\mathcal A},a>$$ on Banach n-tuples.
Starting from n operator ideals $${\mathcal A}_ i$$ (quasi-normed operator ideals $$<{\mathcal A}_ i,a_ i>)$$ on Banach spaces, $$i=1,...,n$$, the authors construct an operator ideal $${\mathcal A}$$ on Banach n-tuples (a quasi-normed operator ideal $$<{\mathcal A},a>$$ on Banach n-tuples). In the second part the usual definitions of products and quotients of (quasi- normed) operator ideals on Banach spaces are extended to (quasi-normed) operator ideals on Banach 2-tuples. So its no surprise that the authors can prove the following theorem:
let $$0<p$$, $$q<1$$, if $$<{\mathcal A},a>$$ is a p-normed operator ideal on Banach 2-tuples, $$<{\mathcal B},b>$$ is a q-normed operator ideal on Banach 2- tuples, then the product $$<{\mathcal A},a>\circ <{\mathcal B},b>$$ and the quotient $$<{\mathcal A},a>^{-1}\circ <{\mathcal B},b>$$, are r-normed operator ideals, where $$1/r:=(1/p+1/q).$$
Finally, for a given quasi-normed operator ideal $$<{\mathcal A},a>$$ the authors state that the minimal kernel of $$<{\mathcal A},a>$$, i.e. $<{\mathcal A}_{\min},a_{\min}>:=<{\mathcal S},\| \cdot \| >\circ <{\mathcal A},a>\circ <{\mathcal S}\quad,\| \cdot \| >,$ and the maximal hull of $$<{\mathcal A},a>$$, i.e. $<{\mathcal A}_{\max},a_{\max}>:=<{\mathcal S},\| \cdot \| >^{-1}\circ <{\mathcal A},a>\circ <{\mathcal S},\| \cdot \| >^{-1},$ are quasi-normed operator ideals, where $$<{\mathcal S},\| \cdot \| >$$ is the operator ideal of the class of all approximable operators.
Reviewer: U.Grimmer
##### MSC:
 47L10 Algebras of operators on Banach spaces and other topological linear spaces