We obtain a new natural description of the class of radial weights for which some previous results of A. Harutyunyan and W. Lusky concerning the boundedness of differentiation and integration operators on corresponding spaces are valid. To do this, we develop an elementary approach which is essentially different from the previous one and allows us to establish several new results and new characterizations of some popular classes of radial weights.

We study weighted composition operators acting between Fock spaces. The following results are obtained:(i) Criteria for the boundedness and compactness.(ii) Characterizations of compact differences and essential norm.(iii) Complete descriptions of path connected components and isolated points of the space of composition operators and the space of nonzero weighted composition operators.

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1. In order to enlarge the class of L. Schwartz's distributions, many authors developed several ultradistributions theories. We only mention A. Beurling [4], G. Björck [5], R. Braun, R. Meise and B.A. Taylor [6], J. Ciorȃnescu and L. Zsidó [7], H. Komatsu [10], J.L. Lions and E. Magenes [11], C. Roumieu [13,14]. Each ultradistribution theory is based on a family (D α ) α∈A consisting of spaces of test functions and satisfying some formal assumptions (see [7, Section 7]). If (D α ) α∈A and (D β ) β∈B are theories of ultradistributions, then (D α ) α∈A is called larger than (D β ) β∈B if for each β ∈ B there exists α ∈ A such that D α ⊂ D β . The two theories are equivalent if each one is larger than the other. By [7] Roumieu-Komatsu theory is strictly larger than Beurling-Björck one while Ciorȃnescu-Zsidó and RoumieuKomatsu theories are equivalent in one-dimensional case. In [6] it has been shown that Braun-Meise-Taylor theory is equivalent to Roumieu-Komatsu one in N-dimensional case and extends Ciorȃnescu-Zsidó theory to R N . In this connec-* Corresponding author. E-mail address: abanin@math.rsu.ru (A.V. Abanin).

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