1 Introduction and Notation

The origin of absolutely p-summing operators can be found in the works of Grothendieck, although the person who formally introduces the basic concepts and makes a systematic study of them is Pietsch (1968). In his monograph (Pietsch 1980), which is considered a basic text for anyone interested in the subject, we can find an extensive treatise on operator ideals that unifies the theory, not only of absolutely p-summing operators, but of a large amount of ideals of operators and their relationships.

Since the inception of the theory, absolutely p-summing operators have found multiple applications. For example, Pelczynski (1966) proved that for Hilbert spaces, the class of all Hilbert-Schmidt operators coincides with the class of all absolutely p-summing operators for arbitrary p, \(1\le p<\infty \). Rosenthal also used absolutely p-summing operators to prove that for \(1\le p<2\) every closed subspace of \(L_p\) either contains a complemented isomorph of \(\ell _p\), or is isomorphic to a subspace of \(L_q\) for some \(q>p\); in particular, every reflexive subspace of \(L_1\) is isomorphic to a subspace of \(L_p\), for some \(p>1\).

The classical book by Diestel et al. (1995) offers an accessible overview of the theory of absolutely p-summing operators, from its foundations to some of its most profound results. It is worth mentioning Grothendieck’s Résumé (Grothendieck 1956), which is one of the most valued works in Functional Analysis, where the power of tensor products in the theory of Banach spaces is shown. Tensor products have turned out to be a very effective tool for the study of operator ideals. Indeed, it is well known that a linear operator \(T:E\rightarrow F\) defined between Banach spaces E and F is absolutely 1-summing if, and only if, the operator \(id\otimes T:\ell _1\otimes _\epsilon E\rightarrow \ell _1\otimes _\pi F \) is continuous, where \(\epsilon \) and \(\pi \) are the injective and projective tensor norms respectively and \(id\otimes T\) is given by \(id\otimes T(\sum _{i=1}^n e_i\otimes x_i)=\sum _{i=1}^n e_i\otimes T(x_i)\), being \(e_i\) the canonical unit vectors and \(x_i\in E\), \(i=1,\ldots ,n\). However, when considering \(1<p<\infty \) the continuity of \(id\otimes T:\ell _p\otimes _\epsilon E\rightarrow \ell _p\otimes _\pi F \) characterizes the p-nuclear operators as introduced by Cohen (1973), whereas T is absolutely p-summing if, and only if, \(id\otimes F: \ell _p\otimes _\epsilon E\rightarrow \ell _p\otimes _{\Delta _p} F\) is continuous, where \(\Delta _p\) satisfies \(\Delta _p(\sum _{i=1}^n e_i\otimes x_i)=(\sum _{i=1}^n\Vert x_i\Vert ^p)^{1/p}\). A unified tensor product point of view based on the above continuities for multiideals can be found in Rueda et al. (2017). The study of operator ideals from the point of view of tensor products is carried out in the monograph by Defant and Floret (1993).

Operator ideals and, in particular, absolutely summing operators have had a profound impact on Functional Analysis. Indeed, in the last few decades a large number of mathematicians have extended the theory of operator ideals beyond its original context related to the linear operators between Banach spaces. One of the most studied scenarios is that of multilinear operators, which was initiated by Pietsch (1983) and that has led to the introduction of several multilinear, polynomial and even Lipschitz extensions of the concept of absolutely p-summing operators (see e.g. Angulo-López and Fernández-Unzueta 2020; Bombal et al. 2004; Dimant 2003; Matos 2003; Pellegrino et al. 2016). The interest for the non-linear theory is also due to its deep connection with important multilinear and polynomial inequalities (Bohnenblust–Hille, Hardy–Littlewood, Kahane–Salem–Zygmund, for instance) and their applications in different fields such as Computer Science. Some of the most recent and significant works in these regards are, for example, Albuquerque and Rezende (2021), Arunachalam et al. (2021), Bayart (2022), Botelho and Wood (2022), Pellegrino and Raposo (2022) and Raposo and Serrano-Rodríguez (2023), where you can also find an extensive list of important references on the subject.

The vast theory of operator ideals has been extended in two main directions. On the one hand, by considering ideals of non necessarily linear mappings as mentioned above, such as multilinear mappings (ideals of multilinear operators or multiideals), m-homogeneous polynomials (ideals of polynomials), Lipschitz mappings (ideals of Lipchitz operators), holomorphic mappings..., and the different ways of generating such non linear operator ideals as well as the study of their properties. On the other hand, when considering classes determined by common properties that characterize the type of linear or non linear operators, as the class of compact, weakly compact, absolutely p-summing, p-nuclear, multiple p-summing, factorable p-summing, completely continuous, strictly singular, absolutely continuous or strongly p-summing linear or non linear operators, among many other classes. Of course, these two aspects of the theory are inseparable and intersect in their development. It is worth pointing out that the leap from studying ideals to multiideals or polynomial ideals often leads to important differences in the properties that satisfy these ideals, giving rise to unexpected situations.

Among the most studied operator ideals we can find the absolutely p-summing operators introduced by Pietsch, the strongly p-summing operators or the p-nuclear operators introduced by Cohen. These ideals are strongly related. For instance, every p-nuclear operator is absolutely p-summing and strongly p-summing, whereas the composition of an absolutely p-summing operator with a strongly p-summing one gives a p-nuclear operator (see [Cohen 1973, Theorem 2.2.1] and the definitions therein). Strongly q-summing operators and absolutely p-summing operators are deeply related via duality whenever \(\frac{1}{p}+\frac{1}{q}=1\) (see [Cohen 1973, Theorem 2.2.2]).

Since Pietsch’s seminal paper (Pietsch 1983) on summing multilinear operators, several different summabilities have appeared in the nonlinear context as we have mentioned before. The respective classes of summing multilinear operators are, in general, proper classes of bounded multilinear operators. At the same time that the theory of nonlinear summing operators was consolidated, intermediate classes of operators appeared that interpolated these summing classes with the entire class of bounded operators.

The first of these interpolating classes was formed by the \((p,\sigma )\)-absolutely continuous multilinear operators, for \(1\le p<\infty \) and \(0\le \sigma <1\), introduced by Matter (1987) in the linear case (see also Pellegrino et al. 2012; López Molina and Sánchez-Pérez 2000) and extended to the multilinear setting by Dahia et al. (2013). This class interpolates in some how the class of absolutely summing operators and the class of bounded linear operators, and it has been conceived as a useful tool when dealing with weaker summabilities properties than that given by the extreme case of absolutely p-summing operators, that merge whenever \(\sigma =0\).

Another example of an interpolating class can be found in Achour et al. (2014), where the class of strongly \((p,\sigma )\)-continuous operators has been studied and extended to the multilinear setting. The extreme case \(\sigma =0\) corresponds to the space of strongly p-summing operators as introduced by Cohen in the linear case (Cohen 1973).

Our aim is to consider an intermediate class of summing multilinear operators in between the ideal of all bounded multilinear operators and the ideal of all Cohen p-nuclear multilinear operators (see Definition 2). This new class is formed by what we call \(\left( p_1,\ldots ,p_m,\sigma ,q,\nu \right) \)-nuclear m -linear mappings, that reduces to a natural multilinear extension of the classical p-nuclear linear operators introduced by Cohen in his seminal paper (Cohen 1973), whenever \(\sigma =\nu =0\). This multilinear extension that merges for \(\sigma =\nu =0\) has been considered and studied in Achour and Alouani (2010). The linear counterpart of this intermediate multilinear class of nuclear operators was studied by Dahia et al. (2021). A variant of nuclear multilinear operators can be found in Achour et al. (2018), where polynomials/multilinear mappings whose linearizations are p-nuclear were considered with the aim of improving the connection with the linear theory.

In Sect. 2, once the new class has been introduced, we use the concept of abstract summing operator to provide a domination theorem for \(\left( p_1,\ldots ,p_m,\sigma ,q,\nu \right) \)-nuclear m-linear mappings. The first notion of abstract summing operator was given by Botelho et al. in Botelho et al. (2010), where a unifying version of the classical Pietsch domination theorem recovered several known results for a number of classes of mappings that generalize the ideal of absolutely p-summing linear operators. The hypotheses of this abstract version were relaxed by Pellegrino and Santos (2011). Finally, in Pellegrino et al. (2012) a full abstract version was provided that recovers more general situations. As an application, we relate our new class with some other classes of multilinear operators, as the class of \((p;p_1,\ldots ,p_m;\sigma )\)-absolutely continuous m-linear operators, the class of strongly \((p,\sigma )\)-continuous m-linear operators or the class of compact operators.

The last section is devoted to the study of the duality of the space of all \(\left( p_1,\ldots ,p_m,\sigma ,q,\nu \right) \)-nuclear m-linear mappings \(T:X_1\times \cdots \times X_m\longrightarrow Y\), that is denoted by \({\mathcal {N}}_{p_1,\ldots ,p_m,\sigma ,q,\nu }^{m}(X_1,\ldots ,X_m;Y)\) and endowed with the nuclear norm. This duality is given in terms of tensor products. In concrete we prove that \({\mathcal {N}}_{p_1,\ldots ,p_m,\sigma ,q,\nu }^{m}(X_1,\ldots ,X_m;Y^*)\) is isometrically isomorphic to the dual of the tensor product \(X_1\otimes \cdots \otimes X_m\otimes Y\) when endowed with a certain reasonable crossnorm that we introduce.

We use standard Banach space notation. Let m be a positive integer and let \(X,X_{1},\ldots ,X_{m},Y\) be Banach spaces over \({\mathbb {K}}= {\mathbb {R}}\) or \({\mathbb {C}}\). For \(1\le p\le \infty , \) \(p^{*}\) denotes the conjugate of p,  i.e., \(\frac{1}{p}+\frac{1}{p^{*}}=1.\) We write \( X^{*}\) to denote the topological dual of X. The closed unit ball of X is represented by \(B_{X}\).

The Banach space of all continuous m-linear operators from \(X_{1}\times \cdots \times X_{m}\) into Y endowed with the supremum norm is denoted by \({\mathcal {L}}\left( X_{1},\ldots ,X_{m};Y\right) .\) In the case \(Y={\mathbb {K}}\) we write \({\mathcal {L}}(X_{1},\ldots ,X_{m})\). When \(m=1\) the space \({\mathcal {L}}\left( X_{1};Y\right) \) reduces to the space of all bounded linear operators from \(X_1\) to Y.

The duality theory in the multilinear frame is strongly related to tensor products. Let us recall some basic notions as we shall make use of them in Sect. 3. The projective norm on the m-fold tensor product \(X_1\otimes \cdots \otimes X_m\) is defined by

$$\begin{aligned} \pi \left( u\right) =\inf \left\{ \overset{n}{\underset{i=1}{\sum }}\underset{j=1}{\overset{m}{\prod }}\left\| x_{i}^{j}\right\| ,\text { }n\in {\mathbb {N}},\text { }u=\overset{n}{\underset{i=1}{\sum }}x_{i}^{1}\otimes \cdots \otimes x_{i}^{m}\right\} . \end{aligned}$$

where the infimum is taken over all representations of u of the form \(\sum \nolimits _{i=1}^{n} x_{i}^{1}\otimes \cdots \otimes x_{i}^{m}.\) The completion of \(X_{1}\otimes \cdots \otimes X_{m}\) when endowed with the projective tensor norm is denoted by \(X_{1}{\widehat{\otimes }}_{\pi }\cdots {\widehat{\otimes }}_{\pi }X_{m}\).

For \(1\le p<\infty \), let us define the classical sequence spaces we shall work with:

\(\bullet \) \(\ell _{p}(X) =\) absolutely p-summable X-valued sequences with the usual norm \(\Vert \cdot \Vert _{p}\);

\(\bullet \) \(\ell _{p,w}\left( X\right) =\) weakly p-summable X-valued sequences with the norm

$$\begin{aligned} \Vert (x_{i})_{i=1}^{\infty }\Vert _{p,w} =\left\| (x_{i})_{i=1}^{\infty }\right\| _{\ell _{p,w}\left( X\right) }=\underset{\varphi \in B_{X^{*}}}{\sup }\left( \overset{\infty }{\underset{i=1}{\sum }}\left| \left\langle x_{i},\varphi \right\rangle \right| ^{p}\right) ^{\frac{1}{p }}; \end{aligned}$$

\(\bullet \) The \((p,\sigma )\)-weakly summable sequences were introduced in López Molina and Sánchez-Pérez (1993) in order to give a characterization of the class of \((p,\sigma )\)-absolutely continuous operators. Now we recall the basics of \((p,\sigma )\)-weak summability.

Fix \(\left( x_{i}\right) _{i=1}^{n}\) in X. Let \(1\le p<\infty \) and \( 0\le \sigma <1\), we define

$$\begin{aligned} \delta _{p\sigma }\left( \left( x_{i}\right) _{i=1}^{n}\right) =\underset{ x^{*}\in B_{X^{*}}}{\sup }\left( \overset{n}{\underset{i=1}{\sum }} \left( \left| \left\langle x_{i},x^{*}\right\rangle \right| ^{1-\sigma }\left\| x_{i}\right\| ^{\sigma }\right) ^{\frac{p}{ 1-\sigma }}\right) ^{\frac{1-\sigma }{p}}. \end{aligned}$$

Note that

$$\begin{aligned} \delta _{p\sigma }(x_1',\ldots ,x_{n'}',x_1'',\ldots ,x_{n''}'')^{\frac{p}{1-\sigma }}\le \delta _{p\sigma }\left( (x_i')_{i=1}^{n'}\right) ^{\frac{p}{1-\sigma }}+\delta _{p\sigma }\left( (x_i'')_{i=1}^{n''}\right) ^{\frac{p}{1-\sigma }}.\nonumber \\ \end{aligned}$$
(1)

For the extreme cases \(\sigma =1\) and \(p=\infty \), we define also for all \( 0\le \tau \le 1\) and \(1\le \eta \le \infty \)

$$\begin{aligned} \delta _{\eta 1}\left( \left( x_{i}\right) _{i=1}^{n}\right) =\delta _{\infty \tau }\left( \left( x_{i}\right) _{i=1}^{n}\right) =\underset{1\le i\le n}{\sup }\left\| x_{i}\right\| =\left\| \left( x_{i}\right) _{i=1}^{n}\right\| _{\infty }. \end{aligned}$$

It is clear that

$$\begin{aligned} \left\| \left( x_{i}\right) _{i=1}^{n}\right\| _{\frac{p}{1-\sigma },w}\le \delta _{p\sigma }\left( \left( x_{i}\right) _{i=1}^{n}\right) \le \left\| \left( x_{i}\right) _{i=1}^{n}\right\| _{\frac{p}{1-\sigma } }. \end{aligned}$$
(2)

The following notion of ideal of multilinear mappings (multi-ideals) goes back to Pietsch (1983).

Definition 1

An ideal of multilinear mappings (or multi-ideal) is a subclass \({\mathcal {M}}\) of all continuous multilinear mappings between Banach spaces such that for all \(m\in {\mathbb {N}}\) and all Banach spaces \(X_{1},\ldots ,X_{m}\) and Y,  the components \({\mathcal {M}}(X_{1},\ldots ,X_{m};Y):={\mathcal {L}} (X_{1},\ldots ,X_{m};Y)\cap {\mathcal {M}}\) satisfy:

(i) \({\mathcal {M}}(X_{1},\ldots ,X_{m};Y)\) is a linear subspace of \(\mathcal {L }(X_{1},\ldots ,X_{m};Y)\) which contains the m-linear mappings of finite type.

(ii) The ideal property: If \(T\in {\mathcal {M}}(G_{1},\ldots ,G_{m};F),u_{j} \in {\mathcal {L}}(X_{j};G_{j})\) for \(j=1,\ldots ,m\) and \(v\in {\mathcal {L}}(F;Y)\), then \(v\circ T\circ (u_{1},\ldots ,u_{m})\) is in \({\mathcal {M}} (X_{1},\ldots ,X_{m};Y).\)

Furthermore, if there is a function \(\left\| .\right\| _{{\mathcal {M}}}:\mathcal { M\longrightarrow }{\mathbb {R}}^{+}\) satisfying

\(\left( 1\right) \) \(\left( {\mathcal {M}}(X_{1},\ldots ,X_{m};Y),\left\| .\right\| _{{\mathcal {M}}}\right) \) is a normed (Banach) space for all Banach spaces \(X_{1},\ldots ,X_{m}\) and Y and,

\(\left( 2\right) \) \(\left\| T^{m}:{\mathbb {K}}^{m}\longrightarrow {\mathbb {K}}:T^{m}\left( x^{1},\ldots ,x^{m}\right) =x^{1}\cdots x^{m}\right\| _{ {\mathcal {M}}}=1\) for all m, 

(3) If \(T\in {\mathcal {M}}(G_{1},\ldots ,G_{m};F),\) \(u_{j}\in {\mathcal {L}} (X_{j};G_{j})\) for \(j=1,\ldots ,m\) and \(v\in {\mathcal {L}}(F;Y),\) then \( \left\| v\circ T\circ (u_{1},\ldots ,u_{m})\right\| _{{\mathcal {M}}}\le \left\| v\right\| \left\| T\right\| _{{\mathcal {M}}}\left\| u_{1}\right\| \cdots \left\| u_{m}\right\| ,\)

then \(({\mathcal {M}},\left\| .\right\| _{{\mathcal {M}}})\) is called a normed (Banach) multi-ideal.

We begin by presenting different classes of ideals of multilinear mappings related to the concept of absolutely summing operator:

\(\bullet \) Let \((\Pi _{p},\pi _{p})\) be the Banach ideal of p-absolutely summing linear operators, for \(1\le p<\infty \).

\(\bullet \) Let \(\left( \Pi _{p,\sigma },\pi _{p,\sigma }\right) \) be the Banach ideal of \((p,\sigma )\)-absolutely continuous linear operators, for \(0\le \sigma \le 1\) and \(1\le p\le \infty \).

The notion of \(\left( p,\sigma \right) \) -absolutely continuous operators was introduced by Matter (1987). For \(1\le p <\infty \), a linear operator \(T\in {\mathcal {L}}(X,Y)\) is \(\left( p,\sigma \right) \) -absolutely continuous if there exist a Banach space G and an operator \(S\in \Pi _{p}\left( X,G\right) \) such that

$$\begin{aligned} \left\| T(x)\right\| \le \left\| x\right\| ^{\sigma }\left\| Sx\right\| ^{1-\sigma }, \ \ x\in X. \end{aligned}$$
(3)

In this case, we write \(\pi _{p,\sigma }=\inf \pi _{p}\left( S\right) ^{1-\sigma },\) where the infimum is taken over all Banach spaces G and all \(S\in \Pi _{p}\left( X,G\right) \) such that (3) holds. Clearly \(\Pi _{p,0}\) coincides with the ideal \(\Pi _{p}.\) If \(0\le \sigma \le 1\) and \(1\le p\le \infty ,\) we put \(\Pi _{\infty ,\sigma }=\Pi _{p,1}= {\mathcal {L}}\) (see Sánchez Pérez 1994).

\(\bullet \) Let \(({\mathcal {L}}_{as(p;p_{1},\ldots ,p_{m})}^{\sigma }, \left\| .\right\| _{{\mathcal {L}}_{as(p;p_{1},\ldots ,p_{m})}^\sigma })\) be the Banach ideal of all \((p;p_{1},\ldots ,p_{m};\sigma ) \) -absolutely continuous m-linear operators, for \(1\le p;p_{1},\ldots ,p_{m}<\infty \) with \(\frac{1}{p}\le \frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}\) and \(0\le \sigma <1.\) This is an extension of the notion of \((p,\sigma )\)-absolutely summing linear operators to the multilinear setting.

A mapping \( T\in {\mathcal {L}}(X_{1},\ldots ,X_{m};Y)\) is \((p;p_{1},\ldots ,p_{m};\sigma ) \) -absolutely continuous (Achour et al. 2013, Definition 1.1), if there is a constant \( C>0\) such that

$$\begin{aligned} \left\| \left( T\left( x_{i}^{1},\ldots ,x_{i}^{m}\right) \right) _{i=1}^{n}\right\| _{\frac{p}{1-\sigma }}\le C\underset{j=1}{\overset{m}{ \prod }}\delta _{p_{j}\sigma }\left( \left( x_{i}^{j}\right) _{i=1}^{n}\right) \end{aligned}$$
(4)

for all choices of \(m\in {\mathbb {N}}\) and \(x_{1}^{j}, \ldots ,x_{n}^{j}\in X_{j},\) \( 1\le j\le m \). The space of all such m-linear operators is denoted by \({\mathcal {L}} _{as(p;p_{1},\ldots ,p_{m})}^{\sigma }(X_{1},\ldots ,X_{m};Y)\) and is endowed with the norm given by

$$\begin{aligned} \left\| T\right\| _{{\mathcal {L}}_{as(p;p_{1},\ldots ,p_{m})}^{\sigma }}=\inf \left\{ C>0:C\text { satisfies }( 4) \right\} . \end{aligned}$$

In the case that \(p_{1}=\cdots =p_{m}=q\) and \(\frac{1}{p}=\frac{1}{p_{1}} +\cdots +\frac{1}{p_{m}}\) we say that T is \(\left( q,\sigma \right) \) -dominated continuous and we denote the corresponding vector space and norm by \({\mathcal {L}}_{d,q}^{\sigma }(X_{1},\ldots ,X_{m};Y)\) and \(\left\| .\right\| _{{\mathcal {L}}_{d,q}^{\sigma }}\) respectively. In this case, the inequality (4) can be written as

$$\begin{aligned} \left\| \left( T\left( x_{i}^{1},\ldots ,x_{i}^{m}\right) \right) _{i=1}^{n}\right\| _{\frac{p}{m\left( 1-\sigma \right) }}\le C\underset{ j=1}{\overset{m}{\prod }}\delta _{q\sigma }\left( \left( x_{i}^{j}\right) _{i=1}^{n}\right) . \end{aligned}$$

It is well known (see Dahia et al. 2013, Theorem 3.3) that T is \((p;p_{1},\ldots ,p_{m};\sigma )\)-absolutely continuous multilinear mapping if, and only if, there is a constant \(C>0\) and a regular Borel probability measure \(\mu _{j}\) on \(B_{X_{j}^{*}},1\le j\le m,\) (with the weak star topology) so that for all \(\left( x^{1},\ldots ,x^{m}\right) \in X_{1}\times \cdots \times X_{m}\), we have

$$\begin{aligned} \left\| T\left( x^{1},\ldots ,x^{m}\right) \right\| \le C\underset{j=1}{\overset{m}{\prod }}\left( \int _{B_{X_{j}^{*}}}\left( \left| \phi \left( x^{j}\right) \right| ^{1-\sigma }\left\| x^{j}\right\| ^{\sigma }\right) ^{\frac{p_{j}}{1-\sigma }}d\mu _{j}\left( \phi \right) \right) ^{\frac{1-\sigma }{p_{j}}}. \end{aligned}$$

\(\bullet \) Let \(1\le p,r<\infty \) and \(0\le \sigma <1\) be such that \(\frac{1}{ r}+\frac{1-\sigma }{p^{*}}=1.\) An m-linear mapping \(T\in {\mathcal {L}} (X_{1},\ldots ,X_{m};Y)\) is strongly \((p,\sigma )\)-continuous if there is a constant \(C>0\) such that for any \(x_{1}^{j},\ldots ,x_{n}^{j}\in X_{j},1\le j\le m\) and any \(y_{1}^{*},\ldots ,y_{n}^{*}\in Y^{*},\) we have

$$\begin{aligned} \left\| \left( \left\langle T\left( x_{1}^{j},\ldots ,x_{n}^{j}\right) ,y_{i}^{*}\right\rangle \right) _{i=1}^{n}\right\| _{1}\le C\left( \underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{m}{\prod }} \left\| x_{i}^{j}\right\| ^{r}\right) ^{\frac{1}{r}}\delta _{p^{*}\sigma }\left( \left( y_{i}^{*}\right) _{i=1}^{n}\right) . \end{aligned}$$
(5)

The collection of all strongly \((p,\sigma )\)-continuous m-linear maps \( X_{1}\times \cdots \times X_{m}\rightarrow Y\) is denoted \({\mathcal {D}} _{p}^{m,\sigma }(X_{1},\ldots ,X_{m};Y).\) The least C for which (5) holds is written \(\left\| T\right\| _{{\mathcal {D}}_{p}^{m,\sigma }}.\) This defines a norm for the space \( {\mathcal {D}}_{p}^{m,\sigma }(X_{1},\ldots ,X_{m};Y).\) The definition of strongly \((p,\sigma )\)-continuous m-linear operator is due to Achour et al. in Achour et al. (2014). Clearly \({\mathcal {D}} _{p}^{m,0}\) coincides with the ideal \({\mathcal {D}}_{p}^{m}\) of Cohen strongly p-summing m-linear operators (see Achour and Mezrag 2007).

\(\bullet \) According to [17], a linear operator \(T\in {\mathcal {L}}(X,Y)\) is \(( p,\sigma ,q,\nu )\)-nuclear (\(1<p,q<\infty \) and \(0\le \sigma ,\nu <1\) such that \(\frac{1-\sigma }{p}+\frac{1-\nu }{q}=1)\), if there exists a constant \(C>0\) such that for every \(\left( x_{i}\right) _{i=1}^{n}\subset X \) and \(\left( y_{i}^{*}\right) _{i=1}^{n}\subset Y^{*}\) the following inequality holds

$$\begin{aligned} \left\| \left( \left\langle T\left( x_{i}\right) ,y_{i}^{*}\right\rangle \right) _{i=1}^{n}\right\| _{1}\le C\delta _{p\sigma }\left( \left( x_{i}\right) _{i=1}^{n}\right) \delta _{q\nu }\left( \left( y_{i}^{*}\right) _{i=1}^{n}\right) . \end{aligned}$$
(6)

The class of all \(\left( p,\sigma ,q,\nu \right) \)-nuclear operators between X and Y is denoted by \({\mathcal {N}}_{p,\sigma ,q,\nu }\left( X,Y\right) .\) The infimum of all the constant C in the inequality (6) defines a norm on \( {\mathcal {N}}_{p,\sigma ,q,\nu }\left( X,Y\right) \) denoted by \({\mathcal {N}}_{p,q}^{\sigma ,\nu }.\)

2 The Multi-ideal of Nuclear Multilinear Operators

In this section we extend the concept of \(\left( p,\sigma ,q,\nu \right) \)-nuclear linear operator to multilinear mappings. The resulting multi-ideal \({\mathcal {N}}_{p_1,\ldots ,p_m,\sigma ,q,\nu }^{m}\) of all \(\left( p_1,\ldots ,p_m,\sigma ,q,\nu \right) \)-nuclear multilinear operators is a normed (Banach) m-ideal. We prove a domination theorem for such operators that will allow to relate this new class of multilinear operators with former related classes, as the class of strongly \((p,\sigma )\)-continuous multilinear mappings, the class of \((q,\sigma )\)-dominated continuous multilinear mappings or the class of compact operators.

Definition 2

Let \(1<p_1,\ldots ,p_m,q<\infty \) and \(0\le \sigma ,\nu <1\) be such that \(\frac{ 1-\sigma }{p_1}+\cdots +\frac{1-\sigma }{p_m}+\frac{1-\nu }{q}=1.\) An m-linear mapping \(T:X_{1}\times \cdots \times X_{m}\rightarrow Y\) is said to be \(\left( p_1,\ldots ,p_m,\sigma ,q,\nu \right) \)-nuclear if there exists a constant \(C>0\) such that for any \(n\in {\mathbb {N}}\), any \(x_{1}^{j},\ldots ,x_{n}^{j}\in X_{j}\), \(1\le j\le m\), and any \(y_{1}^{*},\ldots ,y_{n}^{*}\in Y^{*}\), we have

$$\begin{aligned} \left\| \left( \left\langle T\left( x_{i}^{1},\ldots ,x_{i}^{m}\right) ,y_{i}^{*}\right\rangle \right) _{i=1}^{n}\right\| _{1}\le C\underset{j=1}{\overset{m}{\prod }}\delta _{p_j\sigma }\left( \left( x_{i}^j\right) _{i=1}^{n}\right) \delta _{q\nu }\left( \left( y_{i}^{*}\right) _{i=1}^{n}\right) . \end{aligned}$$

The collection of all \(\left( p_1,\ldots ,p_m,\sigma ,q,\nu \right) \)-nuclear m-linear mappings from \(X_{1}\times \cdots \times X_{m}\) to Y is denoted by \({\mathcal {N}}_{p_1,\ldots ,p_m,\sigma ,q,\nu }^{m}\left( X_{1},\ldots ,X_{m};Y\right) . \) We use \(\left\| T\right\| _{ {\mathcal {N}}_{p_1,\ldots ,p_m,q}^{m\left( \sigma ,\nu \right) }}\) to denote the least \(C>0\) satisfying the above inequality. This determines a norm for the space \({\mathcal {N}}_{p_1,\ldots ,p_m,\sigma ,q,\nu }^{m}\left( X_{1},\ldots ,X_{m};Y\right) .\) It is easy to check that any \(T\in {\mathcal {N}} _{p_1,\ldots ,p_m,\sigma ,q,\nu }^{m}\left( X_{1},\ldots ,X_{m};Y\right) \) is continuous and

$$\begin{aligned} \left\| T\right\| \le \left\| T\right\| _{{\mathcal {N}} _{p_1,\ldots ,p_m,q}^{m\left( \sigma ,\nu \right) }}. \end{aligned}$$

\({\mathcal {N}}_{p_1,\ldots ,p_m,\sigma ,q,\nu }^{m}\) is a Banach multi-ideal with the norm \(\left\| .\right\| _{{\mathcal {N}}_{p_1,\ldots ,p_m,q}^{m \left( \sigma ,\nu \right) }}.\) If \(p_1=\cdots =p_m=p\), we just write for short \({\mathcal {N}}_{p,\sigma ,q,\nu }^{m}\).

The ideal \({\mathcal {N}}_{p_1,\ldots ,p_m,\sigma ,q,\nu }^{m}\) can be considered as a multilinear extension of the class of \((p,\sigma ,q,\nu )\)-nuclear linear operators defined in [17].

If \(\sigma =\nu =0\), \({\mathcal {N}}_{p_1,\ldots ,p_m,0,q,0 }^{m}\) determines the class of multilinear operators \(T:X_1 \times \cdots \times X_m\longrightarrow Y\) that satisfy the following inequality

$$\begin{aligned}{} & {} \left\| \left( \left\langle T\left( x_{i}^{1},\ldots ,x_{i}^{m}\right) ,y_{i}^{*}\right\rangle \right) _{i=1}^{n}\right\| _{1}\nonumber \\{} & {} \le \left\| T\right\| _{{\mathcal {N}}_{p_1,\ldots ,p_m,q}^{m\left( 0,0 \right) }}\underset{j=1}{\overset{m}{\prod }}\Vert ( x_{i}^j) _{i=1}^{n}\Vert _{\ell _{p_j,w}(X_j)} \Vert \left( y_{i}^{*}\right) _{i=1}^{n}\Vert _{\ell _{q,w}(Y^*)}. \end{aligned}$$
(7)

Note that this definition apparently differs from the one given in Achour and Alouani (2010) concerning the space \({\mathcal {N}} _{p}^{m}\left( X_{1},\ldots ,X_{m};Y\right) \) of Cohen p -nuclear m-linear operators, even when \(p_1=\cdots =p_m\) and \(q=p^*\). However, a simple comparison of Theorem 1 with (Achour and Alouani 2010, Theorem 2.5) gives that the inequality (7) also characterizes Cohen p -nuclear m-linear operators when \(p_1=\cdots =p_m\) and \(q=p^*\).

We start by proving the following domination theorem. For its proof we use the full general Pietsch domination theorem presented by Pellegrino et al. (2012).

Theorem 1

An m-linear operator \(T\in {\mathcal {L}}\left( X_{1},\ldots ,X_{m};Y\right) \) is \(\left( p_1,\ldots ,p_m,\sigma ,q,\nu \right) \)-nuclear if, and only if, there exists a positive constant \(C>0\) , Borel probability measures \(\mu _{j}\) on \(B_{X_{j}^{*}}\) \( \left( 1\le j\le m\right) \) and \(\lambda \) on \( B_{Y^{**}}\) such that for all \(\left( x^{1},\ldots ,x^{m}\right) \in X_{1}\times \cdots \times X_{m}\)and \( y^{*}\in Y^{*}\)

$$\begin{aligned} \left| \left\langle T(x^{1},\ldots ,x^{m}),y^{*}\right\rangle \right|\le & {} C\underset{j=1}{\overset{m}{\prod }}\left\| x^{j}\right\| ^{\sigma }\left( \int _{B_{X_{j}^{*}}} \left| \left\langle x^{j},\varphi ^{j}\right\rangle \right| ^{p_j}d\mu \left( \varphi ^{j}\right) \right) ^{\frac{1-\sigma }{p_j}} \\{} & {} \cdot \left\| y^{*}\right\| ^{\nu }\left( \int _{B_{Y^{**}}} \left| \left\langle y^{*},\varphi \right\rangle \right| ^{q}d\lambda \left( \varphi \right) \right) ^{\frac{1-\nu }{q}}. \end{aligned}$$

Proof

Any \(\left( p_1,\ldots ,p_m,\sigma ,q,\nu \right) \)-nuclear m-linear operator T is \(R_{1},\ldots ,R_{m+1}\)-S-abstract \(\left( p_{1},\ldots ,p_{m+1}\right) \) -summing (see [Pellegrino et al. 2012, Definition 4.4]) for the parameters

\(\left\{ \begin{array}{l} t=m+1 \\ r=1 \\ \mathcal {H=L}(X_{1},\ldots ,X_{m};Y) \\ E_1 = {\mathbb {R}} \\ G_j = X_j, ~\text {for all}~ j \in \{1,\ldots ,m\} \\ G_{m+1} = Y^* \\ K_{j}=B_{X_{j}^{*}}~\text {equipped with the weak star topology}, ~\text { for all}~ j \in \{1,\ldots ,m\} \\ K_{m+1}=B_{Y^{**}}~\text {equipped with the weak star topology} \\ p_{j}=\frac{p_j}{1-\sigma }~\text {with}~ 1\le j\le m \\ p_{m+1}= \frac{q}{1-\nu } \\ S: {\mathcal {L}}(X_1,\ldots ,X_m;Y)\times {\mathbb {R}}\times X_1\times \cdots \times X_m\times Y^*\rightarrow [0,\infty ), \\ S(T,\lambda ,x^{1},\ldots ,x^{m},y^*)= \left| \left\langle T(x^{1},\ldots ,x^{m}),y^{*}\right\rangle \right| \\ R_j:B_{X_j^*}\times {\mathbb {R}}\times X_j\rightarrow [0,\infty ), R_{j}\left( \varphi ,\lambda ,x^j\right) = \left| \left\langle x^j,\varphi \right\rangle \right| ^{1-\sigma }\Vert x^j\Vert ^{\sigma }, 1\le j\le m \\ R_{m+1}:B_{Y^{**}}\times {\mathbb {R}}\times Y^*\rightarrow [0,\infty ), \quad R_{m+1}\left( \varphi ^{**} ,\lambda ,y^{*}\right) = \left| \left\langle y^{*}, \varphi ^{**}\right\rangle \right| ^{1-\nu } \Vert y^{*}\Vert ^\nu . \end{array} \right. \)

Theorem 4.6 in Pellegrino et al. (2012) gives the result. \(\square \)

As a direct corollary we get the following.

Corollary 1

Let \(X_1,\ldots ,X_m,Y\) be Banach spaces. Let \(0\le \sigma ,\nu <1\) and \(1\le p_1,\ldots ,p_m,q,r_1,\ldots ,r_m,s<\infty \) be such that \(\frac{1-\sigma }{p_1}+\cdots +\frac{1-\sigma }{p_m}+\frac{1-\nu }{q}=1\) and \(\frac{1-\sigma }{r_1}+\cdots +\frac{1-\sigma }{r_m}+\frac{1-\nu }{s}=1\). If \(s\le q\) and \(r_j\le p_j\) for all \(j=1,\ldots ,m\), then

$$\begin{aligned} {\mathcal {N}}_{p_1,\ldots ,p_m,\sigma ,q,\nu }^m(X_1,\ldots ,X_m;Y)\subset {\mathcal {N}}_{r_1,\ldots ,r_m,\sigma ,s,\nu }^m(X_1,\ldots ,X_m;Y). \end{aligned}$$

Comparing Theorem 1 with (Dahia et al. 2013, Theorem 3.3) we can relate \((p_1,\ldots ,p_m,\sigma ,q,\nu )\)-nuclearity with \(\left( p;p_1,\ldots ,p_m;\sigma \right) \)-absolute continuity. Given an m-linear mapping \(T:X_1\times \cdots \times X_m\longrightarrow Y\), we consider the associated \((m+1)\)-linear functional \(S:X_1\times \cdots \times X_m\times Y^*\longrightarrow {\mathbb {K}}\) given by \(S(x_1,\ldots ,x_m,y^*)=\langle T(x_1,\ldots ,x_m),y^*\rangle \).

Corollary 2

Let \(X_1,\ldots ,X_m,Y\) be Banach spaces and let \(0\le \sigma <1\) and \(1\le p_1,\ldots ,p_m,q<\infty \) be such that \(\frac{1-\sigma }{p_1}+\cdots +\frac{1-\sigma }{p_m}+\frac{1-\sigma }{q}=1\). Then \(T\in {\mathcal {N}}_{p_1,\ldots ,p_m,\sigma ,q,\sigma }^m(X_1,\ldots ,X_m;Y)\) if, and only if, \(S\in {\mathcal {L}}_{as,(p,p_1,\ldots ,p_m)}^{\sigma }(X_{1},\ldots ,X_{m},Y;{\mathbb {K}})\) for any \(p\ge 1\) such that \(\frac{1}{p}\le \frac{1}{p_1}+\cdots + \frac{1}{p_m}+\frac{1}{q}\). Moreover, \(\left\| T\right\| _{{\mathcal {N}} _{p_1,\ldots ,p_m,q}^{m\left( \sigma ,\sigma \right) }}=\Vert S\Vert _{{\mathcal {L}}_{as(p;p_1,\ldots ,p_m,q)}^\sigma }\).

Another application of Theorem 1 is that it allows the class of \((p_1,\ldots ,p_m,\sigma ,q,\nu )\)-nuclear operators to be related to other known classes.

Theorem 2

Let \(1<p_1,\ldots ,p_m,q<\infty \) and \(0\le \sigma ,\nu <1\) be such that \(\frac{1-\sigma }{p_1}+\cdots +\frac{1-\sigma }{p_m}+\frac{1-\nu }{q}=1\). Let \(T:X_{1}\times \cdots \times X_{m}\rightarrow Y\) be an m-linear operator.

(1) If \(T\in {\mathcal {N}}_{p_1,\ldots ,p_m,\sigma ,q,\nu }^{m}\left( X_{1},\ldots ,X_{m};Y\right) \), then \(T\in {\mathcal {D}}_{q^{*}}^{m,\nu }\left( X_{1},\ldots ,X_{m};Y\right) ,\) and \(\Vert T\Vert _{{\mathcal {D}}_{q^* }^{m,\nu }}\le \left\| T\right\| _{{\mathcal {N}} _{p_1,\ldots ,p_m,q}^{m\left( \sigma ,\nu \right) }}\).

(2) If \(T\in {\mathcal {N}}_{p_1,\ldots ,p_m,\sigma ,q,\nu }^{m}\left( X_{1},\ldots ,X_{m};Y\right) , \) then \(T\in {\mathcal {L}}_{as,(p,p_1,\ldots ,p_m)}^{\sigma }(X_{1},\ldots ,X_{m};Y),\) for any \(p\ge 1\) such that \(\frac{1}{p}\le \frac{1}{p_1}+\cdots +\frac{1}{p_m},\) and \(\Vert T\Vert _{{\mathcal {L}}_{as(p;p_1,\ldots ,p_m,q)}^\sigma }\le \left\| T\right\| _{{\mathcal {N}} _{p_1,\ldots ,p_m}^{m\left( \sigma ,\nu \right) }}\).

Proof

(1) If T is a \(\left( p_1,\ldots , p_m,\sigma ,q,\upsilon \right) \)-nuclear m-linear operator, then

$$\begin{aligned} \left| \left\langle T(x^{1},\ldots ,x^{m}),y^{*}\right\rangle \right|\le & {} \left\| T\right\| _{{\mathcal {N}}_{p_1,\ldots , p_m,q}^{m\left( \sigma ,\nu \right) }}\underset{j=1}{\overset{m}{\prod }} \left\| x^{j}\right\| ^{\sigma } \left( \int _{B_{X_{j}^{*}}} \left| \left\langle x^{j},\varphi ^{j}\right\rangle \right| ^{p_j}d\mu \left( \varphi ^{j}\right) \right) ^{\frac{1-\sigma }{p_j}} \\{} & {} \cdot \left\| y^{*}\right\| ^{\nu }\left( \int _{B_{Y^{**}}} \left| \left\langle y^{*},\varphi \right\rangle \right| ^{q}d\lambda \left( \varphi \right) \right) ^{\frac{1-\nu }{q}} \\\le & {} \left\| T\right\| _{{\mathcal {N}}_{p_1,\ldots , p_m,q}^{m\left( \sigma ,\nu \right) }}\underset{j=1}{\overset{m}{\prod }}\left\| x^{j}\right\| ^{\sigma }\underset{\varphi ^{j}\in B_{X_{j}^{*}}}{\sup }\left| \left\langle x^{j},\varphi ^{j}\right\rangle \right| ^{1-\sigma } \\{} & {} \cdot \left\| y^{*}\right\| ^{\nu }\left( \int _{B_{Y^{**}}} \left| \left\langle y^{*},\varphi \right\rangle \right| ^{q}d\lambda \left( \varphi \right) \right) ^{\frac{1-\nu }{q}} \\= & {} \left\| T\right\| _{{\mathcal {N}}_{p_1,\ldots , p_m,q}^{m\left( \sigma ,\nu \right) }}\underset{j=1}{\overset{m}{\prod }}\left\| x^{j}\right\| \left\| y^{*}\right\| ^{\nu }\left( \int _{B_{Y^{**}}} \left| \left\langle y^{*},\varphi \right\rangle \right| ^{q}d\lambda \left( \varphi \right) \right) ^{\frac{1-\nu }{q}}. \\ \end{aligned}$$

By Pietsch’s domination theorem for strongly \((q^{*},\nu )\)-continuous multilinear operators (Achour et al. 2014, Theorem 4.3), \(T\in {\mathcal {D}}_{q^{*}}^{m,\nu }\left( X_{1},\ldots ,X_{m};Y\right) \) and \(\left\| T\right\| _{{\mathcal {D}}_{q^{*}}^{m,\upsilon }}\le \left\| T\right\| _{{\mathcal {N}}_{p_1,\ldots , p_m,q}^{m\left( \sigma ,\nu \right) }}.\)

(2) Take \(p\ge 1\) be such that \(\frac{1}{p}\le \frac{1}{p_1}+\cdots +\frac{1}{p_m}\). If T is a \(\left( p_1,\ldots , p_m,\sigma ,q,\nu \right) \)-nuclear m -linear operator, then

$$\begin{aligned} \left\| T(x^{1},\ldots ,x^{m})\right\|= & {} \underset{y^{*}\in B_{Y^{*}}}{\sup }\left| \left\langle T(x^{1},\ldots ,x^{m}),y^{*}\right\rangle \right| \\\le & {} \left\| T\right\| _{{\mathcal {N}}_{p_1,\ldots ,p_m,q}^{m\left( \sigma ,\nu \right) }}\underset{j=1}{\overset{m}{\prod }}\left\| x^{j}\right\| ^{\sigma }\left( \int _{B_{X_{j}^{*}}} \left| \left\langle x^{j},\varphi ^{j}\right\rangle \right| ^{p_j}d\mu \left( \varphi ^{j}\right) \right) ^{\frac{1-\sigma }{p_j}} \\{} & {} \cdot \underset{y^{*}\in B_{Y^{*}}}{\sup } \left\| y^{*}\right\| ^{\nu }\left( \int _{B_{Y^{**}}} \left| \left\langle y^{*},\varphi \right\rangle \right| ^{q}d\lambda \left( \varphi \right) \right) ^{ \frac{ 1-\nu }{q}} \\\le & {} \left\| T\right\| _{{\mathcal {N}}_{p_1,\ldots ,p_m,q}^{m\left( \sigma ,\nu \right) }}\underset{j=1}{\overset{m}{\prod }}\left\| x^{j}\right\| ^{\sigma }\left( \int _{B_{X_{j}^{*}}}\left| \left\langle x^{j},\varphi ^{j}\right\rangle \right| ^{p_j}d\mu \left( \varphi ^{j}\right) \right) ^{\frac{1-\sigma }{p_j}} \\{} & {} \cdot \underset{y^{*}\in B_{Y^{*}}}{\sup } \underset{\varphi \in B_{Y^{**}}}{\sup }\left| \left\langle y^{*},\varphi \right\rangle \right| ^{1-\nu }\left\| y^{*}\right\| ^{\nu } \\\le & {} \left\| T\right\| _{{\mathcal {N}}_{p_1,\ldots ,p_m,q}^{m\left( \sigma ,\nu \right) }}\underset{j=1}{\overset{m}{\prod }}\left\| x^{j}\right\| ^{\sigma }\left( \int _{B_{X_{j}^{*}}}\left| \left\langle x^{j},\varphi ^{j}\right\rangle \right| ^{p_j}d\mu \left( \varphi ^{j}\right) \right) ^{\frac{1-\sigma }{p_j}}. \end{aligned}$$

Therefore by Pietsch’s domination theorem for \(\left( p;p_1,\ldots ,p_m;\sigma \right) \) -absolutely continuous m-linear operators (Dahia et al. 2013, Theorem 3.3), \(T\in {\mathcal {L}}_{as,(p,p_1,\ldots ,p_m)}^{\sigma }(X_{1},\ldots ,X_{m};Y)\) and \(\left\| T\right\| _{{\mathcal {L}}_{as,(p,p_1,\ldots ,p_m)}^{\sigma }}\le \left\| T\right\| _{ {\mathcal {N}}_{p_1,\ldots ,p_m,q}^{m\left( \sigma ,\nu \right) }}.\) \(\square \)

Corollary 3

Let \(1<p_1,\ldots ,p_m,q<\infty \) and \(0\le \sigma ,\nu <1\) be such that \(\frac{1-\sigma }{p_1}+\cdots +\frac{1-\sigma }{p_m}+\frac{1-\nu }{q}=1\). Let Y be a Banach space and \(X_{1},\ldots ,X_{m}\) be reflexive Banach spaces. If \( T\in {\mathcal {N}}_{p_1,\ldots ,p_m,\sigma ,q,\nu }^{m}\left( X_{1},\ldots ,X_{m};Y\right) \), then T is compact.

Proof

By Theorem 2, the operator T is \(\left( p;p_1,\ldots ,p_m;\sigma \right) \) -absolutely continuous, and has a reflexive domain. Then T is compact (see [Dahia et al. 2013, Corollary 5.2.]). \(\square \)

3 \({\mathcal {N}}_{p_{1},\ldots ,p_{m},\sigma ,q,\nu }^{m}\left( X_{1},\ldots ,X_{m};Y\right) \) as a dual space

The aim of this section is to introduce a reasonable crossnorm on the tensor product \( X_{1}\otimes \cdots \otimes X_{m}\otimes Y\) so that the topological dual of the resulting space is isometrically isomorphic to \(\left( {\mathcal {N}}_{p_{1},\ldots ,p_{m},\sigma ,q,\nu }^{m}\left( X_{1},\ldots ,X_{m};Y\right) ,\left\| .\right\| _{{\mathcal {N}}_{p_{1},\ldots ,p_{m},q}^{m\left( \sigma ,\nu \right) }}\right) \).

For \(1<p_{1},\ldots ,p_{m},q<\infty \) and \(0\le \sigma ,\nu <1\) such that \(\frac{1-\sigma }{p_{1}}+\cdots + \frac{ 1-\sigma }{p_{m}}+\frac{1-\nu }{q}=1\) and for \(u\in X_{1}\otimes \cdots \otimes X_{m}\otimes Y,\) we consider

$$\begin{aligned} g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\left( u\right) =\inf \underset{j=1}{\overset{m}{\prod }}\delta _{p_{j}\sigma }\left( ( x_{i}^j) _{i=1}^{n}\right) \delta _{q\nu }\left( \left( y_{i}\right) _{i=1}^{n}\right) , \end{aligned}$$
(8)

where the infimum is taken over all representations of u of the form \(u={{\sum _{i=1}^{n} }}x_{i}^{1}\otimes \cdots \otimes x_{i}^{m}\otimes y_{i},\) with \(x_{i}^{j}\in X_{j},y_{i}\in Y,1\le i\le n,1\le j\le m\) and \({n\in }{\mathbb {N}}.\)

The next result gives the basic properties of \(g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\). Recall that the injective tensor norm \(\epsilon \) on \(X_{1}\otimes \cdots \otimes X_{m}\otimes Y\) is given by

$$\begin{aligned} \epsilon (u)=\sup \left\{ \sum _{i=1}^n x_1^*(x_i^1)\cdots x_m^*(x_i^m)y^*(y_i): y^*\in B_{Y^*}, x_j^*\in B_{X_j^*},1\le j\le m\right\} , \end{aligned}$$

for any \(u=\sum \nolimits _{i=1}^nx_i^1\otimes \cdots \otimes x_i^m\otimes y_i\in X_{1}\otimes \cdots \otimes X_{m}\otimes Y\).

Proposition 3

\(g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\) is a reasonable crossnorm and \(\epsilon \le g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }.\)

Proof

We start by proving the triangle inequality. Let \(u,v\in \) \(X_{1}\otimes \cdots \otimes X_{m}\otimes Y,\) and let \( \varepsilon >0\). Choose representations of u and v of the form

$$\begin{aligned} u=\underset{i=1}{\overset{n^{\prime }}{\sum }}x_{i}^{\prime 1}\otimes \cdot \cdot \cdot \otimes x_{i}^{\prime m}\otimes y_{i}^{\prime }\text { and }v= \underset{i=1}{\overset{n^{\prime \prime }}{\sum }}x_{i}^{\prime \prime 1}\otimes \cdots \otimes x_{i}^{\prime \prime m}\otimes y_{i}^{\prime \prime } \end{aligned}$$

such that

$$\begin{aligned} \underset{j=1}{\overset{m}{\prod }}\delta _{p_{j}\sigma }\left( \left( x_{i}^{\prime j }\right) _{i=1}^{n^{\prime }}\right) \delta _{q\nu }\left( \left( y_{i}^{\prime }\right) _{i=1}^{n^{\prime }}\right) \le g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\left( u\right) +\varepsilon \end{aligned}$$

and

$$\begin{aligned} \underset{j=1}{\overset{m}{\prod }}\delta _{p_{j}\sigma }\left( \left( x_{i}^{\prime \prime j }\right) _{i=1}^{n^{\prime \prime }}\right) \delta _{q\nu }\left( \left( y_{i}^{\prime \prime }\right) _{i=1}^{n^{\prime \prime }}\right) \le g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\left( v\right) +\varepsilon . \end{aligned}$$

We can write u, v in the following way

$$\begin{aligned} u=\underset{i=1}{\overset{n^{\prime }}{\sum }}z_{i}^{\prime 1}\otimes \cdot \cdot \cdot \otimes z_{i}^{\prime m}\otimes t_{i}^{\prime }\text { and }v= \underset{i=1}{\overset{n^{\prime \prime }}{\sum }}z_{i}^{\prime \prime 1}\otimes \cdots \otimes z_{i}^{\prime \prime m}\otimes t_{i}^{\prime \prime } \end{aligned}$$

with

$$\begin{aligned} z_{i}^{\prime j}= & {} \dfrac{\left( g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\left( u\right) +\varepsilon \right) ^{\frac{1-\sigma }{ p_j}}}{\delta _{p_{j}\sigma }\left( \left( x_{k}^{\prime j}\right) _{k=1}^{n^{\prime }}\right) }x_{i}^{\prime j},\ \ \ j=1,\ldots ,m,\text { } i=1,\ldots ,n^{\prime }, \\ t_{i}^{\prime }= & {} \dfrac{\prod _{j=1}^{m}\delta _{p_{j}\sigma }\left( \left( x_k^{\prime j}\right) _{k=1}^{n^{\prime }}\right) }{\left( g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\left( u\right) +\varepsilon \right) ^{1-\frac{1-\nu }{q}}} y_{i}^{\prime },\text { }i=1,\ldots ,n^{\prime }, \\ z_{i}^{\prime \prime j}= & {} \dfrac{\left( g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\left( v\right) +\varepsilon \right) ^{\frac{ 1-\sigma }{p_j}}}{\delta _{p_{j}\sigma }\left( \left( x_k^{\prime \prime }\right) _{k=1}^{n^{\prime \prime }}\right) }x_{i}^{\prime \prime j},\ \ \ j=1,\ldots ,m,\text { }i=1,\ldots ,n^{\prime \prime }, \\ t_{i}^{\prime \prime }= & {} \dfrac{\underset{j=1}{\overset{m}{\prod }}\delta _{p_{j}\sigma }\left( \left( x_k^{\prime \prime j}\right) _{k=1}^{n^{\prime \prime }}\right) }{\left( g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\left( v\right) +\varepsilon \right) ^{1-\frac{1-\nu }{ q}}}y_{i}^{\prime \prime },\text { }i=1,\ldots ,n^{\prime \prime }. \end{aligned}$$

It follows that

\(\delta _{p_{j}\sigma }\left( \left( z_{i}^{\prime j}\right) _{i=1}^{n^{\prime }}\right) =\left( g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\left( u\right) +\varepsilon \right) ^{\frac{ 1-\sigma }{p_j}}\) and \(\delta _{q\nu }\left( \left( t_{i}^{\prime }\right) _{i=1}^{n^{\prime }}\right) \le \left( g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\left( u\right) +\varepsilon \right) ^{\frac{ 1-\nu }{q}}\)

\(\delta _{p_{j}\sigma }\left( \left( z_{i}^{\prime \prime j}\right) _{i=1}^{n^{\prime }}\right) =\left( g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\left( v\right) +\varepsilon \right) ^{\frac{ 1-\sigma }{p_j}}\) and \(\delta _{q\nu }\left( \left( t_{i}^{\prime \prime }\right) _{i=1}^{n^{\prime \prime }}\right) \le \left( g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\left( v\right) +\varepsilon \right) ^{\frac{1-\nu }{q}}.\)

Using the inequality (1) we get that

$$\begin{aligned} g_{( p_1,\ldots ,p_m,q) }^{( \sigma ,\nu ) }( u+v)\le & {} \prod _{j=1}^m\left( \left( \delta _{p_j\sigma }(z_i^{\prime j})_{i=1}^{n'}\right) ^{\frac{p_j}{1-\sigma }}+\left( \delta _{p_j\sigma }(z_i^{\prime \prime j})_{i=1}^{n''}\right) ^{\frac{p_j}{1-\sigma }}\right) ^{\frac{1-\sigma }{p_j}}\\{} & {} \cdot \left( \left( \delta _{q,\nu }(t_i')_{i=1}^{n'}\right) ^{\frac{q}{1-\nu }}+\left( \delta _{q,\nu }(t_i'')_{i=1}^{n''}\right) ^{\frac{q}{1-\nu }}\right) ^{\frac{1-\nu }{q}}\\\le & {} \prod _{J=1}^m\left( g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }(u)+g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }(v)+2\epsilon \right) ^{\frac{1-\sigma }{p_j}}\\{} & {} \cdot \left( g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }(u)+g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }(v)+2\epsilon \right) ^{\frac{1-\nu }{q}}.\\ \end{aligned}$$

Letting \(\epsilon \) tend to 0 we get

$$\begin{aligned} g_{( p_1,\ldots ,p_m,q) }^{( \sigma ,\nu ) }(u+v)\le & {} \prod _{j=1}^m\left( g_{( p_1,\ldots ,p_m,q) }^{( \sigma ,\nu ) }(u)+g_{( p_1,\ldots ,p_m,q) }^{( \sigma ,\nu ) }(v)\right) ^{\frac{1-\sigma }{p_j}}\\{} & {} \cdot \left( g_{( p_1,\ldots ,p_m,q) }^{( \sigma ,\nu ) }(u)+g_{( p_1,\ldots ,p_m,q) }^{( \sigma ,\nu ) }(v)\right) ^{\frac{1-\nu }{q}}\\= & {} g_{( p_1,\ldots ,p_m,q) }^{( \sigma ,\nu ) }( u) +g_{( p_1,\ldots ,p_m,q) }^{( \sigma ,\nu ) }( v). \end{aligned}$$

Trivially \( g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\left( \lambda u\right) =\left| \lambda \right| g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\left( u\right) \) for all \(\lambda \in {\mathbb {K}}.\)

Let \(u=\sum \nolimits _{i=1}^{n} x_{i}^{1}\otimes \cdots \otimes x_{i}^{m}\otimes y_{i}\in \) \(X_{1}\otimes \cdots \otimes X_{m}\otimes Y.\) Applying Hölder’s inequality and (2) we get

$$\begin{aligned} \epsilon \left( u\right)= & {} \sup \left\{ \left| \underset{i=1}{\overset{n}{\sum }\ }\varphi ^{1}\left( x_{i}^{1}\right) \cdots \varphi ^{m}\left( x_{i}^{m}\right) \psi \left( y_{i}\right) \right| :\psi \in B_{Y^{*}}, \varphi ^{j}\in B_{X_{j}^{*}},1\le j\le m\right\} \\\le & {} \underset{j=1}{\overset{m}{\prod }} \underset{\varphi ^{j}\in B_{X_{j}^{*}}}{\sup }\left\| \left( \varphi ^{j}\left( x_{i}^{j}\right) \right) _{i=1}^{n}\right\| _{\frac{p_{j}}{1-\sigma } } \underset{\psi \in B_{Y^{*}}}{\sup }\left\| \left( \psi \left( y_{i}\right) \right) _{i=1}^{n}\right\| _{\frac{q}{1-\nu } } \\= & {} \underset{j=1}{\overset{m}{\prod }}\left\| \left( x_{i}^{j}\right) _{i=1}^{n}\right\| _{\frac{p_{j}}{1-\sigma },w}\left\| \left( y_{i}\right) _{i=1}^{n}\right\| _{\frac{q}{1-\nu },w} \le \underset{j=1}{\overset{m}{\prod }}\delta _{p_{j}\sigma }\left( \left( x_{i}^{j}\right) _{i=1}^{n}\right) \delta _{q\nu }\left( \left( y_{i}\right) _{i=1}^{n}\right) . \end{aligned}$$

Taking the infimum over all representations of u we get \(\epsilon \left( u\right) \le g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\left( u\right) .\) Therefore \(g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\left( u\right) =0\) implies \(u=0.\) We have proved that \( g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\) is a norm on \(X_{1}\otimes \cdots \otimes X_{m}\otimes Y\) and that \(\epsilon \le g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\).

Since for every \(x^j\in X_j\), \(j=1,\ldots ,m\), and \(y\in Y\)

$$\begin{aligned} g_{( p_{1},\ldots ,p_{m},q) }^{( \sigma ,\nu ) }( x^1\otimes \cdots \otimes x^m\otimes y)\le \prod _{j=1}^m \delta _{p_j\sigma }(x^j)\delta _{q\nu }(y)=\Vert x^1\Vert \cdots \Vert x^m\Vert \Vert y\Vert , \end{aligned}$$

it easily follows that \(g_{( p_1,\ldots ,p_m,q) }^{( \sigma ,\nu ) }(u)\le \pi (u)\) for every \(u\in X_1\otimes \cdots \otimes X_m\otimes Y\). Hence, \(g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\) is a reasonable crossnorm.\(\square \)

Let us see that the tensor product of continuous linear operators is continuous when the norm \(g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\) is considered. Let \(X,X_{j},Y,Y_{j}\) be Banach spaces, and \(T\in {\mathcal {L}}\left( X,Y\right) ,T_{j}\in {\mathcal {L}}\left( X_{j},Y_{j}\right) ,j=1,\ldots ,m.\) Recall that the linear mapping \(T_1\otimes \cdots \otimes T_m\otimes T:X_1\otimes \cdots \otimes X_m\otimes X\rightarrow Y_1\otimes \cdots \otimes Y_m\otimes Y\) is given by

$$\begin{aligned} (T_1\otimes \cdots \otimes T_m\otimes T)(x_1\otimes \cdots \otimes x_m\otimes x)=T_1(x_1)\otimes \cdots \otimes T_m(x_m)\otimes T(x). \end{aligned}$$

Proposition 4

Let \(X,X_{j},Y,Y_{j}\) be Banach spaces, and \(T\in {\mathcal {L}}\left( X,Y\right) ,T_{j}\in {\mathcal {L}}\left( X_{j},Y_{j}\right) ,j=1,\ldots ,m.\) Then, the linear operator \(T_1\otimes \cdots \otimes T_m\otimes T:(X_1\otimes \cdots \otimes X_m\otimes X,g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }) \rightarrow (Y_1\otimes \cdots \otimes Y_m\otimes Y,g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) })\) is continuous. Moreover,

$$\begin{aligned} \left\| T_{1}\otimes \cdots \otimes T_{m}\otimes T\right\| =\left\| T\right\| \underset{j=1}{\overset{m}{\prod }}\left\| T_{j}\right\| . \end{aligned}$$

Proof

We may assume that \(T_{j}\ne 0,j=1,\ldots ,m\) and \(T\ne 0.\) Let \(u=\sum \nolimits _{i=1}^{n} x_{i}^{1}\otimes \cdots \otimes x_{i}^{m}\otimes x_{i}\in X_{1} \otimes \cdots \otimes X_{m}\otimes X\). Then,

$$\begin{aligned} \underset{i=1}{\overset{n}{\sum }}\left( T_{1}x_{i}^{1}\right) \otimes \cdots \otimes \left( T_{m}x_{i}^{m}\right) \otimes \left( Tx_{i}\right) \end{aligned}$$

is a representation of \(T_{1}\otimes \cdots \otimes T_{m}\otimes T\left( u\right) \) in \(Y_{1}\otimes \cdots \otimes Y_{m}\otimes Y.\) Hence,

$$\begin{aligned} g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\left( T_1\otimes \cdots \otimes T_m\otimes T\left( u\right) \right)\le & {} \underset{j=1}{\overset{m}{\prod }}\delta _{p_{j}\sigma }\left( \left( T_{j}x_{i}^{j}\right) _{i=1}^{n}\right) \delta _{q\nu }\left( \left( Tx_{i}\right) _{i=1}^{n}\right) \\\le & {} \left\| T\right\| \underset{j=1}{\overset{m}{\prod }}\left\| T_{j}\right\| \underset{j=1}{\overset{m}{\prod }}\delta _{p_{j}\sigma }\left( \left( x_{i}^{j}\right) _{i=1}^{n}\right) \delta _{q\nu }\left( \left( x_{i}\right) _{i=1}^{n}\right) . \end{aligned}$$

Since this holds for every representation of u,  we obtain

$$\begin{aligned} g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\left( T_{1}\otimes \cdots \otimes T_{m}\otimes T\left( u\right) \right) \le \left\| T\right\| \underset{j=1}{\overset{m}{\prod }}\left\| T_{j}\right\| g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\left( u\right) . \end{aligned}$$

The above inequality gives the continuity and

$$\begin{aligned} \left\| T_{1}\otimes \cdots \otimes T_{m}\otimes T\right\| \le \left\| T\right\| \underset{j=1}{\overset{m}{\prod }}\left\| T_{j}\right\| . \end{aligned}$$
(9)

The reverse inequality follows from Proposition 3. Indeed, as \(g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\) is a reasonable crossnorm, we get that

$$\begin{aligned} \left\| Tx\right\| \underset{j=1}{\overset{m}{\prod }}\left\| T_{j}x^{j}\right\|= & {} g_{\left( p_1,\ldots ,p_m,q\right) }^{\left( \sigma ,\nu \right) }\left( \left( T_{1}x^{1}\right) \otimes \cdots \otimes \left( T_{m}x^{m}\right) \otimes \left( Tx\right) \right) \\\le & {} \left\| T_{1}\otimes \cdots \otimes T_{m}\otimes T\right\| g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\left( x^{1}\otimes \cdots \otimes x^{m}\otimes x\right) \\\le & {} \left\| T_{1}\otimes \cdots \otimes T_{m}\otimes T\right\| \left\| x\right\| \underset{j=1}{\overset{m}{\prod }}\left\| x^{j}\right\| . \end{aligned}$$

Taking suprema on the respective unit balls it follows that

$$\begin{aligned} \left\| T_{1}\otimes \cdots \otimes T_{m}\otimes T\right\| \ge \left\| T\right\| \underset{j=1}{\overset{m}{\prod }}\left\| T_{j}\right\| \end{aligned}$$
(10)

by (9) and (10), we have

$$\begin{aligned} \left\| T_{1}\otimes \cdots \otimes T_{m}\otimes T\right\| =\left\| T\right\| \underset{j=1}{\overset{m}{\prod }}\left\| T_{j}\right\| . \end{aligned}$$

\(\square \)

The following result is the main result of this section.

Theorem 5

The space \(\left( {\mathcal {N}}_{p_{1},\ldots ,p_{m},\sigma ,q,\nu }^{m}\left( X_{1},\ldots ,X_{m};Y^{*}\right) ,\left\| .\right\| _{{\mathcal {N}} _{p_{1},\ldots ,p_{m},q}^{m\left( \sigma ,\nu \right) }}\right) \) is isometrically isomorphic to the dual space \(\left( X_{1}\otimes \cdots \otimes X_{m}\otimes Y,g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\right) ^*\) via the mapping \(\psi \) defined by

$$\begin{aligned} \psi (T) \left( x^{1}\otimes \cdots \otimes x^{m}\otimes y\right) =T\left( x^{1},\ldots ,x^{m}\right) \left( y\right) , \end{aligned}$$

for every \(T\in {\mathcal {N}}_{p_{1},\ldots ,p_{m},\sigma ,q,\nu }^{m}\left( X_{1},\ldots ,X_{m};Y^{*}\right) ,x^{j}\in X_{j},j=1,\ldots ,m\) and \( y\in Y.\)

Proof

Given \(T\in {\mathcal {N}}_{p_{1},\ldots ,p_{m},\sigma ,q,\nu }^{m}\left( X_{1},\ldots ,X_{m};Y^{*}\right) \) and \(u=\sum _{i=1}^nx_i^1\otimes \cdots \otimes x_i^m\otimes y_i\in X_1\otimes \cdots \otimes X_m\otimes Y\), it follows from the nuclearity of T that

$$\begin{aligned} \left| \sum _{i=1}^n \langle T(x_i^1,\ldots ,x_i^m),y_i\rangle \right| \le \Vert T\Vert _{{\mathcal {N}} _{p_{1},\ldots ,p_{m},q}^{m\left( \sigma ,\nu \right) }}\prod _{j=1}^m\delta _{p_j\sigma }\left( (x_i^j)_{i=1}^n\right) \delta _{q\nu }\left( (y_i)_{i=1}^n\right) . \end{aligned}$$

Then, the function \(\psi _{T}\) given by \( \psi _T(u)=\sum \nolimits _{i=1}^nT\left( x_i^{1},\ldots ,x_i^{m}\right) \left( y_i\right) \) satisfies

$$\begin{aligned} \left| \psi _T(u)\right| \le \Vert T\Vert _{{\mathcal {N}} _{p_{1},\ldots ,p_{m},q}^{m\left( \sigma ,\nu \right) }}g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }(u). \end{aligned}$$

Hence, \(\psi _T\in \left( X_{1}\otimes \cdots \otimes X_{m}\otimes Y,g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\right) ^*\) and

$$\begin{aligned} \Vert \psi _T\Vert \le \Vert T\Vert _{{\mathcal {N}} _{p_{1},\ldots ,p_{m},q}^{m\left( \sigma ,\nu \right) }}. \end{aligned}$$

Therefore, the correspondence \(T\rightarrow \psi (T)=\psi _{T}\) is well defined and clearly it is linear and injective.

Let us show its surjectivity. Consider \(\phi \in \left( X_{1}\otimes \cdots \otimes X_{m}\otimes Y,g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\right) ^* \) and define the m-linear mapping \(T\in {\mathcal {L}}\left( X_{1},\ldots ,X_{m};Y^*\right) \) by \(T\left( x^{1},\ldots ,x^{m}\right) \left( y\right) =\phi \left( x^{1}\otimes \cdots \otimes x^{m}\otimes y\right) .\)

If \(x_{i}^{j}\in X_{j},\) \(y_{i}\in Y,1\le i\le n,1\le j\le m\), then

$$\begin{aligned} \left| \overset{n}{\underset{i=1}{\sum }}\left\langle T\left( x_{i}^{1},\ldots ,x_{i}^{m}\right) ,y_{i}\right\rangle \right|= & {} \left| \underset{i=1}{\overset{n}{\sum }}\phi \left( x_{i}^{1}\otimes \cdots \otimes x_{i}^{m}\otimes y_{i}\right) \right| \\= & {} \left| \phi \left( \underset{i=1}{\overset{n}{\sum }}x_{i}^{1}\otimes \cdots \otimes x_{i}^{m}\otimes y_{i}\right) \right| \\\le & {} \left\| \phi \right\| g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\left( \underset{i=1}{\overset{n}{\sum }} x_{i}^{1}\otimes \cdots \otimes x_{i}^{m}\otimes y_{i}\right) \\\le & {} \left\| \phi \right\| \underset{j=1}{\overset{m}{\prod }} \delta _{p_{j}\sigma }\left( ( x_{i}^{j}) _{i=1}^{n}\right) \delta _{q\nu }\left( ( y_{i}) _{i=1}^{n}\right) , \end{aligned}$$

which shows that \(T\in {\mathcal {N}}_{p_{1},\ldots ,p_{m},\sigma ,q,\nu }^{m}\left( X_{1},\ldots ,X_{m};Y^{*}\right) \) and \( \left\| T\right\| _{{\mathcal {N}}_{p_{1},\ldots ,p_{m},q}^{m\left( \sigma ,\nu \right) }}\le \left\| \phi \right\| . \) Since \(\phi =\psi _T\), from both inequalities it follows that

$$\begin{aligned} \left\| \psi _{T}\right\| =\left\| T\right\| _{{\mathcal {N}} _{p_{1},\ldots ,p_{m},q}^{m\left( \sigma ,\nu \right) }} \end{aligned}$$

and the proof is finished.\(\square \)

If we take \(Y={\mathbb {K}},\) then we can identify \( X_{1}\otimes \cdots \otimes X_{m}\otimes {\mathbb {K}}\) with \(X_{1}\otimes \cdots \otimes X_{m}\) via the correspondence \(u=\sum _{i=1}^n x_i^1\otimes \cdots \otimes x_i^m\otimes \lambda _i \leftrightarrow v=\sum _{i=1}^n \lambda _i x_i^1\otimes \cdots \otimes x_i^m\). Note that in this case the norm \(g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\) adopts the following description:

$$\begin{aligned} g_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }(u)=\inf \Vert (\lambda _i)_{i=1}^n\Vert _{\frac{q}{1-\nu }}\prod _{j=1}^m\delta _{p_j\sigma }\left( (x_i^j)_{i=1}^n\right) , \end{aligned}$$

where the infimum is taken over all representations of u of the form \(u=\sum \nolimits _{i=1}^n x_i^1\otimes \cdots \otimes x_i^m\otimes \lambda _i \), \(\lambda _i\in {\mathbb {K}}\), \(x_i^j\in X_j\), \(j=1,\ldots ,m\), \(n\in {\mathbb {N}}\). Indeed,

$$\begin{aligned} \delta _{q,\nu }\left( (\lambda _i)_{i=1}^n\right) =\sup _{|\gamma |\le 1}\left( \sum _{i=1}^n\left( |\lambda _i\gamma |^{1-\nu }|\lambda _i|^\nu \right) ^{\frac{q}{1-\nu }}\right) ^{\frac{1-\nu }{q}}=\left( \sum _{i=1}^n|\lambda _i|^{\frac{q}{1-\nu }}\right) ^{\frac{1-\nu }{q}}. \end{aligned}$$

If we denote \(w_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }(v)=\inf \Vert (\lambda _i)_{i=1}^n\Vert _{\frac{q}{1-\nu }}\prod _{j=1}^m\delta _{p_j\sigma }\left( (x_i^j)_{i=1}^n\right) ,\) where the infimum is taken over all representations of \(v\in X_{1}\otimes \cdots \otimes X_{m}\) of the form \(v=\sum _{i=1}^n \lambda _i x_i^1\otimes \cdots \otimes x_i^m\), \(\lambda _i\in {\mathbb {K}}\), \(x_i^j\in X_j\), \(j=1,\ldots ,m\), \(n\in {\mathbb {N}}\), then Theorem 5 gives the following:

Corollary 4

Let \(X_{1},\ldots ,X_{m}\) be Banach spaces. The space of \(\left( p_1,\ldots ,p_m,\sigma ,q,\nu \right) \)-nuclear multilinear forms \(\left( {\mathcal {N}}_{p_{1},\ldots ,p_{m},\sigma ,q,\nu }^{m}\left( X_{1},\ldots ,X_{m}\right) ,\left\| .\right\| _{{\mathcal {N}}_{p_{1},\ldots ,p_{m},q}^{m\left( \sigma ,\nu \right) }}\right) \) is isometrically isomorphic to \(\left( X_{1}\otimes \cdots \otimes X_{m},w_{\left( p_{1},\ldots ,p_{m},q\right) }^{\left( \sigma ,\nu \right) }\right) ^{*}.\)