Approximation numbers of weighted composition operators

Abstract

We study the approximation numbers of weighted composition operators f → w · (f • ϕ) on the Hardy space H 2 on the unit disc. For general classes of such operators, upper and lower bounds on their approximation numbers are derived. For the special class of weighted lens map composition operators with specific weights, we show how much the weight w can improve the decay rate of the approximation numbers, and give sharp upper and lower bounds. These examples are motivated from applications to the analysis of relative commutants of special inclusions of von Neumann algebras appearing in quantum field theory (Borchers triples).

Some examples of composition operators and their approximation numbers on the Hardy space of the bidisk

“…We see in this section that the presence of a "rapidly decaying" weight allows simpler estimates for the approximation numbers of a corresponding weighted composition operator. Such a study, but a bit different, is made in [14].…”

“…We give here an alternative proof, based on a result of Gunatillake ([9]), this result holding in a wider context. Now, we can give a new proof Theorem 2.5 of [14] as follows. Let a ∈ D be such that w(a) ϕ ′ (a) = 0 (H(D) is a division ring and ϕ ′ ≡ 0, w ≡ 0).…”

Compactification, and beyond, of composition operators on Hardy spaces by weights

“…For w ∈ H 2 , the multiplication operator M w is defined formally by f → wf and the weighted composition operator by f → w (f •ϕ). It is known (see [5] for instance) that twisting C ϕ by some M w can improve its compactness properties, and even its membership in Schatten classes S p or the decay of its approximation numbers [7,Theorem 2.3].…”

Singular Values of Weighted Composition Operators and Second Quantization

“…Intriguingly, the link which we explore between singular values of weighted composition operators and two-dimensional conformal field theory is not the only such connection. Very recently, singular values of these operators were studied (with emphasis on compact operators) in [LLQRP16], in the context of modular nuclearity for Borchers triples, a very different application than the one explored here. The simultaneous appearance of two distinct applications of singular values of weighted composition operators to conformal field theory reveals their importance, and suggests further work aimed at understanding the relationship between these connections.…”