The theory of multi-norms was developed by H. G. Dales and M. E. Polyakov in a memoir that was published in Dissertationes Mathematicae. In that memoir, the notion of ’equivalence’ of multi-norms was defined. In the present memoir, we make a systematic study of when various pairs of multi-norms are mutually equivalent. In particular, we study when (p,q)-multi-norms defined on spaces are equivalent, resolving most cases; we have stronger results in the case where r = 2. We also show that the standard [t]-multi-norm defined on is not equivalent to a (p,q)-multi-norm in most cases, leaving some cases open. We discuss the equivalence of the Hilbert space multi-norm, the (p,q)-multi-norm, and the maximum multi-norm based on a Hilbert space. We calculate the value of some constants that arise. Several results depend on the classical theory of (q,p)-summing operators.
@book{bwmeta1.element.bwnjournal-rm-doi-10_4064-dm498-0-1,
author = {H. G. Dales and M. Daws and H. L. Pham and P. Ramsden},
title = {Equivalence of multi-norms},
series = {GDML\_Books},
year = {2014},
zbl = {1317.46012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm498-0-1}
}
H. G. Dales; M. Daws; H. L. Pham; P. Ramsden. Equivalence of multi-norms. GDML_Books (2014), http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm498-0-1/