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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
47L10 Algebras of operators on Banach spaces and other topological linear spaces