Schur and operator multipliers
Lyudmila Turowska2010, Banach Center Publications
Sign up for access to the world's latest research
Abstract
Schur multipliers were introduced by Schur in the early 20th century and have since then found a considerable number of applications in Analysis and enjoyed an intensive development. Apart from the beauty of the subject in itself, sources of interest in them were connections with Perturbation Theory, Harmonic Analysis, the Theory of Operator Integrals and others. Advances in the quantisation of Schur multipliers were recently made in [29]. The aim of the present article is to summarise a part of the ideas and results in the theory of Schur and operator multipliers. We start with the classical Schur multipliers defined by Schur and their characterisation by Grothendieck, and make our way through measurable multipliers studied by Peller and Spronk, operator multipliers defined by Kissin and Shulman and, finally, multidimensional Schur and operator multipliers developed by Juschenko and the authors. We point out connections of the area with Harmonic Analysis and the Theory of Operator Integrals. 1. Classical Schur multipliers For a Hilbert space H, let B(H) be the collection of all bounded linear operators acting on H equipped with its operator norm • op. We denote by ℓ 2 the Hilbert space of all square summable complex sequences. With an operator A ∈ B(ℓ 2), one can associate a matrix (a i,j) i,j∈N by letting a i,j = (Ae j , e i), where {e i } i∈N is the standard orthonormal basis of ℓ 2. The space M ∞ of all matrices obtained in this way is a subspace of the space M N of all complex matrices indexed by N × N. It is easy to see that the correspondence between B(ℓ 2) and M ∞ is one-to-one. Any function ϕ : N × N → C gives rise to a linear transformation S ϕ acting on M N and given by S ϕ ((a i,j) i,j) = (ϕ(i, j)a i,j) i,j. In other words, S ϕ ((a i,j) i,j) is the entry-wise product of the matrices (ϕ(i, j)) i,j and (a i,j) i,j , often called Schur product. The function ϕ is called a Schur multiplier if S ϕ leaves the subspace M ∞ invariant. We denote by S(N, N) the set of all Schur multipliers. Let ϕ be a Schur multiplier. Then the correspondence between B(ℓ 2) and M ∞ gives rise to a mapping (which we denote in the same way) on B(ℓ 2). We first note that S ϕ is necessarily bounded in the operator norm. This follows from the Closed Graph Theorem; indeed, suppose that A k → 0 and S ϕ (A k) → B in the operator norm, for some elements A k , B ∈ B(ℓ 2),
