Inclusions and coincidences for multiple summing multilinear mappings

Abstract

Using complex interpolation we prove new inclusion and coincidence theorems for multiple (fully) summing multilinear and holomorphic mappings. Among several other results we show that continuous n-linear forms on cotype 2 spaces are multiple (2; q k , ..., q k )-summing, where 2 k−1 < n ≤ 2 k , q 0 = 2 and q k+1 = 2q k 1+q k for k ≥ 0.

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

“…From Corollary 3 we can also improve the results in [18,Corollary 4.6], [2,Theorem 3], [3,Theorem 3.6]. In our result, in item (a) it appears the term "one", while in [2,3,18] it appears the term "all".…”

“…In our result, in item (a) it appears the term "one", while in [2,3,18] it appears the term "all". Further item (b) is new.…”

“…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]). …”

Nonlinear absolutely summing operators revisited

“…If r 1 = · · · = r n = s we just write (r; s), and when r = s we replace (r; r) by r. For n = 1 this concept also coincides with the classical notion of absolutely summing linear operators and, for this reason, we keep the usual notation π (r;s) (T ) instead of T (r;s) for the norm of T. The essence of the notion of multiple summing multilinear operators, for bilinear operators, can also be traced back to [71]. For recent results in the theory of multiple summing operators we refer to [8,22,60,68] and references therein.…”