This paper introduces and systematically develops the theory of multilinear (p, S)-summing operators, providing a unifying framework that simultaneously generalizes several known classes of summing multilinear mappings. Our central result establishes a complete Saphar-type duality theorem: for 1 < p < oo, the space of all (p, S)-summing m-linear operators from E-1 & times; & centerdot; & centerdot; & centerdot; & times; E-m into the dual of a Banach space Z is isometrically isomorphic to the dual of the tensor product (E-1 circle times & centerdot; & centerdot; & centerdot; circle times E-m circle times Z, d(p)(S)(& lowast;)), where d(p)(S)(& lowast;) is a new generalized multilinear Saphar seminorm. We prove that this class forms a Banach ideal of multilinear operators and establish a corresponding Pietsch-type factorization theorem through subspaces of vector-valued Lebesgue spaces. As concrete applications, we recover and clarify the known duality results for multilinear (l(p)(s), l(p))-summing and op-dual (l(p)(s), l(p))-summing operators. Furthermore, we provide a detailed analysis of inclusion relations between these ideals, constructing explicit counter examples that demonstrate the strictness of these inclusions.

We explore the procedure given by left-hand quotients in the context of weighted holomorphic ideals. On the one hand, we show that this procedure does not generate new ideals other than the ideal of weighted holomorphic mappings when considering the left-hand quotients induced by the ideals of p-compact, weakly p-compact, unconditionally p-compact, approximable or right p-nuclear operators with their respective weighted holomorphic ideals. On the other hand, the procedure is of interest when considering other operators ideals as it provides new weighted holomorphic ideals. This is the case of the ideal of Grothendieck weighted holomorphic mappings or the ideal of Rosenthal weighted holomorphic mappings, where the applicability of this construction is shown.

In this paper, we introduce and systematically study a new class of linear operators called Cohen-Lorentz $((r,s);(p,q))$-nuclear operators. This concept unifies and generalizes the classical theory of Cohen $(p,q)$-nuclear operators by incorporating the framework of Lorentz sequence spaces. We provide a comprehensive set of equivalent characterizations and establish the ideal properties of this new class. The relationships with other known operator ideals, such as Cohen-Lorentz summing and Cohen-Lorentz strongly summing operators, are investigated in detail. We prove that, under specific conditions on the indices, this class forms a Banach ideal. Furthermore, we provide a tensor product characterization for operators with values in a dual space and establish a fundamental factorization theorem.

Given 1 <= p, q < infinity and 0 <= sigma, eta < 1 such that 1-sigma/ p + 1-eta /q <= 1, we continue the study of the Banach Lipschitz ideal of (p, sigma, q, eta)-dominated operators initiated by Saleh. Among other results, we introduce an associated Lipschitz tensor norm, implying that it is a maximal Banach Lipschitz operator ideal. Also, we show that this ideal extends the Banach operator ideal of (p, sigma, q, eta)-dominated operators. Finally, we exhibit other Lipschitz operator ideals which also extend this operator ideal and we compare them with each other.

In this paper, new developments in the theory of ideals of Bloch maps are utilized to introduce and analyze the properties of $\left(p,\theta,q,\eta\right)$-nuclear Bloch maps from the open unit disk $\mathbb{D}$ to a complex Banach space $X,$ where $1\leq p,q<\infty $ and $0\leq \theta ,\eta<1$ satisfy $\left( 1-\theta \right) /p+\left( 1-\eta \right) /q=1$. The main emphasis is placed on defining these maps, establishing their Banach space properties, and investigating fundamental characteristics such as Pietsch domination, Bloch compactness and Möbius invariance. Finally, we conclude the paper by presenting a Bloch reasonable crossnorm and illustrating the isometric isomorphism between the defined space and its dual space.

Motivated by new progress in the theory of ideals of Bloch maps, we introduce (p,σ ) -absolutely continuous Bloch maps with p∈ [1,∞ ) and σ∈ [0,1) from the complex unit open disc 𝔻 into a complex Banach space X. We prove a Pietsch domination/factorization theorem for such Bloch maps that provides a reformulation of some results on both absolutely continuous (multilinear) operators and Lipschitz operators. We also identify the spaces of (p,σ ) -absolutely continuous Bloch zero-preserving maps from 𝔻 into X^* under a suitable norm π ^ℬ_p,σ with the duals of the spaces of X-valued Bloch molecules on 𝔻 equipped with the Bloch version of the (p^*,σ ) -Chevet–Saphar tensor norms.

Given 1 < p , r < infinity and 0 <= sigma < 1 such that 1/r + (1 - sigma ) / p(& lowast;) = 1, we study the Banach normalized Bloch ideal (D-p,sigma(B)(D, X ) , d(p,sigma)(B)) formed by all strongly (p, sigma )-absolutely continuous Bloch maps from the complex unit open disc D into a complex Banach space X. Characterizations of such Bloch maps are established in terms of: (i) Pietsch domination, (ii) linearisation on G(D) (the Bloch-free Banach space over D), (iii) Bloch transposition, and (iv) Pietsch factorization. The invariance of such maps under Mobius transformations of D and their relation with compact Bloch maps are also addressed. Furthermore, we show that such space can be identified with the dual of the tensor product space lin(Gamma(D))circle times X-& lowast; equipped with a suitable Bloch reasonable crossnorm & rhov;(B)(p,sigma).

The theory of (p, r, s)-summing and (p, r, s)-nuclear linear operators on Banach spaces was developed by Pietsch in his book on operator ideals (Pietsch in Operator ideals, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam, 1980, Chapters 17 and 18) Due to recent advances in the theory of ideals of Bloch maps, we extend these concepts to Bloch maps from the complex open unit disc 𝔻 into a complex Banach space X. Variants for (r, s)-dominated Bloch maps of classical Pietsch’s domination and Kwapień’s factorization theorems of (r, s)-dominated linear operators are presented. We define analogues of Lapresté’s tensor norms on the space of X-valued Bloch molecules on 𝔻 to address the duality of the spaces of (p^*,r,s) -summing Bloch maps from 𝔻 into X^* . The class of (p, r, s)-nuclear Bloch maps is introduced and analysed to give examples of (p, r, s)-summing Bloch maps.

This work aims to define a new class of operators, called strongly ( p,σ) -Lipschitz mappings. We prove a Pietsch type domination/factorization theorem for this class of operators and show it characterizes Lipschitz mappings whose Lipschitz conjugates are absolutely (p^*;σ ) -summing.

Among all the extensions of absolutely summing linear operators to the non linear setting, the class of factorable strongly p-summing multilinear operators or polynomials preserves all the fundamental properties of the linear theory. In this paper we analyse factorable strongly p-summability for non linear operators in the context of Banach lattices and relate it to the linear theory.

In this article, we establish new relationships involving the class of Cohen positive strongly $p$-summing multilinear operators. Furthermore, we introduce a new class of multilinear operators on Banach lattices, called positive Cohen weakly nuclear multilinear operators. We establish a Pietsch domination-type theorem for this new class of multilinear operators. As an application, we show that every positive Cohen weakly $p$-nuclear multilinear operator is positive Dimant strongly $p$-summing and Cohen positive strongly $p$-summing. We conclude with a tensor representation of our class.

This paper introduces the class of ( p_1,… ,p_m,σ ,q,ν) -nuclear m -linear operators between Banach spaces, as an intermediate space between the class of nuclear multilinear operators and the whole class of all bounded multilinear operators. The connection with the theory of summing m-linear operators is established. Moreover, we identify this space with a dual space by means of a reasonable crossnorm inspired by the Chevet–Saphar norm.

Abstract In this paper, we valorize the relationship between positive p−summing operators and positive strongly q−summing operators using (Contemp. Math. 328, 145 − 149 (2003)).

In the present paper, we introduce and investigate a new class of positively p-nuclear operators that are positive analogues of right p-nuclear operators. One of our main results establishes an identification of the dual space of positively p-nuclear operators with the class of positive p-majorizing operators that is a dual notion of positive p-summing operators. As applications, we prove the duality relationships between latticially p-nuclear operators introduced by O. I. Zhukova and positively p-nuclear operators. We also introduce a new concept of positively p-integral operators via positively p-nuclear operators and prove that the inclusion map from $$L_{p^{*}}(\mu )$$ to $$L_{1}(\mu )$$ ( $$\mu $$ finite) is positively p-integral. New characterizations of latticially p-integral operators and positively p-integral operators are presented and used to prove that an operator is latticially p-integral (resp. positively p-integral) precisely when its second adjoint is. Finally, we describe the space of positively p-integral operators as the dual of the $$\Vert \cdot \Vert _{\Upsilon _{p}}$$ -closure of the subspace of finite rank operators in the space of positive p-majorizing operators. Approximation properties, even positive approximation properties, are needed in establishing main identifications.

In this paper, we give and study the notion of positive Cohen p-nuclear operators. We prove a natural analog of the Pietsch domination theorem for these classes, we characterize their conjugates and we valorize some relationship between positive p-summing operators and positive strongly q-summing using (Contemp. Math. 328,145-149(2003)).

In this work we extend the concept of (r; t; s)-nuclear operators presented by Lapresté in (Studia math., T. LVII. 1976, 47 – 83) to n-homogeneous polynomials. Factorization and inclusion properties are described. Under some conditions, we also characterize the topological dual of the studied space.

We introduce the concepts of Cohen positive strongly p -summing and positive p -dominated m-homogeneous polynomials. The version of Pietsch’s domination theorem for the first class among other results and a Bu-type theorem is proved, as well as some inclusions with other known spaces. Moreover, we present a characterization of these classes in tensor terms.

Abstract In this paper, we introduce and study the concept of positive Cohen p-nuclear multilinear operators between Banach lattice spaces. We prove a natural analog to the Pietsch domination theorem for this class. Moreover, we give like the Kwapień’s factorization theorem. Finally, we investigate some relations with another known classes.

In the present paper, we introduce a new concept of positive p-majorizing operators as a dual notion of positive p-summing operators and generalize the concept of majorizing operators introduced by Schaefer (Isr J Math 13:400–415, 1972). We introduce the concept of positive (p, q)-dominated operators and prove a positive version of the famous Kwapień’s factorization theorem for (p, q)-dominated operators via positive p-majorizing operators. We also introduce the notion of disjoint p-summing operators which is a new larger class of operators than positive p-summing operators and use it to characterize the Radon–Nikodým property. Finally, we investigate the maximal properties of these four classes of operators and prove that they are maximal in corresponding sense.

The aim of this paper is to study the onto isometries of the space of strongly Lipschitz p-nuclear operators, introduced by D. Chen and B. Zheng (Nonlinear Anal.,75, 2012). We give some new results about such isometrics and we focus, in particular, on the case F = ℓp*.