Abstract: We characterize all the extreme points of the unit ball in the space of trilinear forms on the Hilbert space C-2. This answers a question posed by R. Grzaslewicz and K. John , who solved the corresponding problem for the real Hilbert space R-2. As an application we determine the best constant in the inequality between the Hilbert-Schmidt norm and the norm of trilinear forms.
Keywords: Trilinear form;
norm; Hilbert-Schmidt norm; extreme point; extreme point of unit ball
Abstract: We describe a new approach to interpolate by the complex method quasi-Banach couples formed by real-intermediate spaces. End-point cases are also considered, and applications are given to function spaces and to operator spaces.
Abstract: The authors consider multiparameter scales of interpolation spaces and prove a general form of the Wolff reiteration theorem [cf. T. H. Wolff, Lecture Notes Math. 908, 199-
204 (1982)] for n- tuples of Banach spaces. The proof, based on the use of the Aronszajn- Gagliardo orbit and co-orbit functors, is an adaptation of previous
work by S. Janson, P. Nilsson, J. Peetre and M. Zafran [Proc. London Math. Soc.,
Abstract: We investigate how compact operators behave under J and K interpolation methods for N spaces and two parameters. First we study those methods: relationship with those already existing in the literature, estimates for the norms of interpolated operators, examples, characterization as Aronszajn-Gagliardo functors,.... We also describe the relationship between Sparr and Fernandez methods
and we derive sharp estimates for the norms of interpolated operators in Fernandez' case. Then we investigate the behaviour of compact operators. We begin with the case when one of the N-tuples reduces to a single Banach space, and later we treat the general case by means of the approach developed in .
Abstract: We give some new examples of bounded multilinear forms on th
Hilbert spaces 2 and L2(0,âˆž). We characterize those which are compact or Hilbert-Schmidt. In particular, we study m-linear forms (m É¥ 3) on 2 which can be regarded as the multilinear analogue of the famous Hilbert matrix. We
Also determine the norm of the permanent on Kn, where K = R or C.
Keywords: Trilinear Forms;
Bilinear-Forms; Matrix; Spectrum; multilinear forms of the type of the Hilbert matrix; distinguished forms on L-2(0, infinity); norm of the permanent;Mathematics
Abstract: In a previous paper, the authors laid the foundations of a theory of Schattenñvon Neumann classes 'p
(0!p%Â¢) of trilinear forms in Hilbert space. This paper continues that research. In the n-dimensional
case, it is shown that the best constant d# that relates the HilbertñSchmidt norm of a form with its bounded
norm behaves like n. Some results are also obtained
in the quasi-Banach case (0!p!1), and for twobounded
forms. Finally, the domination problem is investigated in the trilinear set-up.
Keywords: Two-boundedness; domination; Schatten-von Neumann classes; trilinear forms; quasi-Banach case
Abstract: The authors establish a number of results concerning normed ideals of multilinear forms in Banach spaces which extend the theory of trace ideals of operators on Hilbert space to
such multilinear forms. For example, it is shown that the dual of the space of compact forms is the space of nuclear forms, while the second dual is the space of bounded
forms. Moreover, a direct analogue of the Schatten
p-ideals of operators on Hilbert space is defined using interpolation techniques and a normal form for compact forms is introduced.
Keywords: Schatten p-ideals; normed ideals of multilinear forms in Banach spaces; trace ideals of operators on Hilbert space; compact forms; nuclear forms; interpolation techniques; normal form