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

Abstract

We prove results which show a new distinctive feature between the class of summing, versus dominated and multiple summing operators. We improve also some recent results in this area. Mathematics Subject Classification (2000). Primary 46G25; Secondary 46B25, 46C99.

Sharp coincidences for absolutely summing multilinear operators

“…A result due to G. Botelho [10] asserts that, under certain cotype assumptions, the Defant-Voigt theorem (1.7) can be improved even with arbitrary Banach spaces F in the place of the scalar field K: if n ≥ 2 is a positive integer, s ∈ [2, ∞) and F = {0} is any Banach space, then (1.8) inf r : L(E 1 , ..., E n ; F ) = as(r;1,...,1) (E 1 , ..., E n ; F ) for all E j in C (s) ≤ s n and the value r = s n is attained. Using recent results (see [6,8,49]) it is also simple to conclude that sup r : L(E 1 , ..., E n ; F ) = as(1;r,...,r) (E 1 , ..., E n ; F ) for all E j in C (s) ≥ sn sn + s − n and sup r : L(E 1 , ..., E n ; F ) = as(2;r,...,r) (E 1 , ..., E n ; F ) for all E j in C (s) ≥ 2sn 2sn + s − 2n , with both r = sn sn+s−n and r = 2sn 2sn+s−2n attained. Several recent papers have treated similar problems involving inclusion, coincidence results and the geometry of the Banach spaces involved (see [12,13,29,38,49]).…”

“…Using recent results (see [6,8,49]) it is also simple to conclude that sup r : L(E 1 , ..., E n ; F ) = as(1;r,...,r) (E 1 , ..., E n ; F ) for all E j in C (s) ≥ sn sn + s − n and sup r : L(E 1 , ..., E n ; F ) = as(2;r,...,r) (E 1 , ..., E n ; F ) for all E j in C (s) ≥ 2sn 2sn + s − 2n , with both r = sn sn+s−n and r = 2sn 2sn+s−2n attained. Several recent papers have treated similar problems involving inclusion, coincidence results and the geometry of the Banach spaces involved (see [12,13,29,38,49]). In fact, families of these kind of coincidence results have been obtained by different works and techniques.…”

Composition results for strongly summing and dominated multilinear operators

“…For a comparison between various classes of absolutely summing multilinear operators the reader can consult [30,33,40,42].…”

Nonlinear absolutely summing operators revisited

“…Several indicators from the theory of summing operators and from the theory of (multi-) ideals show that this is one of the most adequate approaches to the nonlinear theory of absolutely summing operators. For results on multiple summing multilinear operators we refer to [10,21,60,62,68,69].…”