Inclusion theorems for absolutely summing holomorphic mappings

Abstract

Abstract. For linear operators, if 1 ≤ p ≤ q < ∞, then every absolutely p-summing operator is also absolutely q-summing. On the other hand, it is well known that for n ≥ 2, there are no general "inclusion theorems" for absolutely summing n-linear mappings or n-homogeneous polynomials. In this paper we deal with situations in which the spaces of absolutely p-summing and absolutely q-summing linear operators coincide, and prove that for 1 ≤ p ≤ q ≤ 2 and n ≥ 2, we have inclusion theorems for absolutely summing n-linear mappings/n-homogeneous polynomials/holomorphic mappings. It is worth mentioning that our results hold precisely in the opposite direction from what is expected in the linear case, i.e., we show that, in some situations, as p increases, the classes of absolutely p-summing mappings becomes smaller.

Nonlinear absolutely summing operators revisited

“…The study of coincidence theorems may follow the lines of [10] combined with the results from the respective linear theories; the study of holomorphic mappings may follow [37] and for inclusion theorems [60] is certainly a good source of inspiration.…”

“…Since then, several authors were attracted by the subject and also non-multilinear approaches have appeared (see [16,17,37,43,45,57]). The adequate way of lifting the notion of a given operator ideal to the multilinear and polynomial settings is a delicate matter.…”

On cotype and a Grothendieck-type theorem for absolutely summing multilinear operators

“…The following result is a combination of [18 The results above are clearly not always optimal since, for example,…”

A new distinguishing feature for summing, versus dominated and multiple summing operators

“…In [17], Pietsch sketched an n-linear approach to the theory of absolutely summing operators and since then a large number of papers has followed this line, e.g., [1][2][3][5][6][7][9][10][11][13][14][15]18], where some extensions of the linear case to the multilinear one are proven. In this paper we give a new distinctive feature between summing versus dominated and multiple summing operators in the multilinear case, concerning the multilinear version of the Pietsch composition theorem.…”

“…We recall some possible extensions of the concept of p-summing operator in the multilinear case (see [1][2][3]7,[9][10][11][13][14][15]18]). …”