A W-transform-based criterion for the existence of bounded extensions of E-operators ✩

Let Γ (H) be the symmetric Fock space over a Hilbert space H and ε :H →Γ (H) the exponential mapping. By an E-operator we mean an operator defined on ε(H). For an E-operator A, the composition mapping Φ = A ◦ ε is called its W-transform. In this paper, we obtain a criterion based on the W-transform for checking whether or not an E-operator becomes a bounded linear operator on the Fock space.  2003 Elsevier Inc. All rights reserved.