Weighted composition operators between Fock spaces in several variables

Abstract

We obtain criteria for the boundedness and compactness of weighted composition operators between different Fock spaces in . We also give estimates for essential norm of these operators. K E Y W O R D S essential norm, Fock space, weighted composition operator M S C ( 2 0 1 0 ) Primary: 47B33; Secondary: 32A15 1200

Topological Structure of the Space of Composition Operators Between Different Fock Spaces

“…The composition operator C ϕ induced by ϕ is defined as C ϕ f = f • ϕ. The boundedness and compactness of composition operators C ϕ : F p (C n ) → F q (C n ) with p, q ∈ (0, ∞) were characterized in [18,Corollaries 3.11 and 3.14]. Here we extend these results for p, q ∈ (0, ∞] and summarize these characterizations in the following theorems.…”

“…∈ L q (C n , d A), which by the same arguments in [18,Corollary 3.14], is equivalent to that ϕ(z) = Az + b with A < 1. Thus, (i) is equivalent to (iii).…”

“…We end this section with the normalization ϕ of ϕ, which was introduced in [5,18] to study boundedness and compactness of (weighted) composition operators on Fock spaces in several variables. Let ϕ(z) = Az + b with an n × n matrix A and an n × 1 vector b.…”

“…Let ϕ(z) = Az + b with an n × n matrix A and an n × 1 vector b. Then, by [10, Theorem 2.6.3], there exist n × n unitary matrices V and U such that A can be written as A = V AU , where A is a diagonal matrix ( a i j ) with [5,18], we call the function ϕ(z) := Az + b with b := V * b a normalization of ϕ with respect to the singular value decomposition A = V AU (briefly, normalization of ϕ).…”