Fully, absolutely summing and Hilbert-Schmidt multilinear mappings.

*(English)*Zbl 1078.46031The 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 \(r=r_1=\dots =r_n \in [2,\infty ]\).

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 \(r=r_1=\dots =r_n \in [2,\infty ]\).

Reviewer: Hans-Andreas Braunß (Potsdam)