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Fully, absolutely summing and Hilbert-Schmidt multilinear mappings. (English) Zbl 1078.46031
The author considers several types of $$n$$-linear mappings defined on the product of Banach spaces $$E_1 \times \ldots \times E_n$$ with values in $$F$$ and their connections among these types.
Fully absolutely $$(r;r_1,\dots ,r_n )$$-summing $$n$$-linear mappings, also known as multiple absolutely $$(r;r_1,\dots ,r_n )$$-summing, are considered in section 2. Several properties and examples are included.
In section 3, the tensor product $$E_1 \otimes \dots \otimes E_n \otimes F$$ is endowed with (quasi-)norms such that its topological dual is isometrically isomorphic to the space of fully absolutely $$(r;r_1,\dots ,r_n )$$-summing $$n$$-linear mappings from $$E_1 \times \dots \times E_n$$ into $$F'$$ if $$r\in [1,\infty ]$$.
In section 4, the author considers virtually $$(r;r_1,\dots ,r_n )$$-nuclear mappings. This class is in general larger than the class of $$(r;r_1,\dots ,r_n )$$-nuclear mappings. It is shown, among other things, that the topological dual of the space of all virtually $$(r;r_1,\dots ,r_n )$$-summing $$n$$-linear mappings on $$E_1 \times \ldots \times E_n$$ into $$F$$ is isometrically isomorphic to the space of all fully absolutely $$(r';r'_1,\ldots ,r'_n )$$-summing $$n$$-linear mappings on $$E'_1 \times \dots \times E'_n$$ into $$F'$$ if the spaces $$E'_1 ,\dots ,E'_n$$ have the bounded approximation property, $$r,r_1,\dots ,r_n \in [1,\infty ]$$ and $$r,r'$$ and $$r_k ,r'_k$$ are conjugate exponents.
In section 5, the space of $$n$$-linear Hilbert-Schmidt mappings between Hilbert spaces is studied. It is shown that this space is isomorphic to the space of fully absolutely $$(r;r_1,\dots ,r_n )$$-summing $$n$$-linear mappings if $$r=r_1=\dots =r_n \in [2,\infty ]$$.

##### MSC:
 46G25 (Spaces of) multilinear mappings, polynomials 47H60 Multilinear and polynomial operators
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