Andrew Tonge

Banach algebras and absolutely summing operators

If R is a Banach algebra and o ∈ R′, the dual space, then we may define a bounded linear map by We shall show that for suitable p the requirement that each be p -absolutely summing constrains R to be an operator algebra, or even, in certain cases, a uniform algebra. In this way we are able to give generalizations of results of Varopoulos (12) and Kaijser (4).

An interpolation approach to Hardy–Littlewood inequalities for norms of operators on sequence spaces

We revisit estimates of Hardy and Littlewood for norms of operators on sequence spaces, and exploit recent advances in interpolation theory to provide relatively simple proofs.

Equivalence constants for matrix norms: a problem of Goldberg

Abstract We investigate the equivalence constants for the l p -coefficient norms and l q -operator norms ( 1⩽p,q⩽∞ ) of complex m×n matrices, and provide estimates which are either best possible or close to best possible.

Polarization and the two-dimensional Grothendieck inequality

Throughout this paper all scalars and vector spaces will be assumed to be complex unless there are specific indications to the contrary.

The failure of Bernstein's theorem for polynomials on C(K)spaces

Abstract Bernsteins theorem asserts that if p: C → C is a polynomial of degree m, then its derivative p′ satisfies the inequality ∥p′∥, ⩽ m ∥p∥, where the symbol | | ∞ denotes the supremum norm taken over the unit disc. Harris [2] proved an analogous inequality for the Frechet derivative of polynomials on Hilbert space. In his commentary to problem 73 in the Scottish Book ( R. D. Mauldin, Ed., Birkhauser, Boston, pp. 144–145, 1981 ), he asked whether there is a similar result for polynomials on C(K) spaces. The purpose of this note is to give a negative answer, even for polynomials of degree 2.

Bilinear Mappings and Trace Class Ideals

In a recent paper [14], Redheffer and Volkmann investigated the norms of bilinear maps (N(cp) • (N(cq)--, Cr. They had been motivated by previous work [12, 13] on a classical result of Schur concerning a special bilinear form on Hilbert space. The purpose of this paper is to sharpen the main results in [14] and to resolve the questions left open in [14]. The new ingredients we use to tackle these problems are interpolation theory for trace class ideals, Neharis theorem on Hankel operators, the Rudin-Shapiro polynomials and a factorisation result of Pisier.

Generalized Hardy inequalities

Abstract We show that the generalized Hardy inequality ∑ k |b k f (n k )|⩽C‖f‖ H 1 holds for f∈H1 and certain (bk)∈lr (2⩽q⩽∞) whenever n k k=1 ∞ ⊂ N satisfies appropriate growth conditions dependent on r.

Random trilinear forms and the Schur multiplication of tensors

We obtain estimates for the distribution of the norm of the random trilinear form A : � M × � N × � K → C, defined by A(ei ,e j ,e k) = aijk, where the aijks are uniformly bounded, independent, mean zero random variables. As an application, we make progress on the problem whenr ˘ ⊗� p ˘ ⊗� q is a Banach algebra under the Schur multiplication.