Let ψ be an entire self-map of the n-dimensional Euclidean complex space C n and u be an entire function on C n . A weighted composition operator induced by ψ with weight u is given by (uC ψ f )(z) = u(z)f (ψ(z)), for z ∈ C n and f is the entire function on C n . In this paper, we study weighted composition operators acting between generalized Fock-types spaces. We characterize the boundedness and compactness of these operators act between F p φ (C n ) and F q φ (C n ) for 0 < p, q ≤ ∞. Moreover, we give estimates for the Fock-norm of uC ψ : F p φ → F q φ when 0 < p, q < ∞, and also when p = ∞ and 0 < q < ∞.

Abstract. Let φ be an analytic self-map of the open unit disk D and g be an analytic function on D. The generalized composition operator induced by the maps g and φ is defined by the integral operatorGiven an admissible weight ω, the weighted Hilbert space H ω consists of all analytic functions f suchIn this paper, we characterize the boundedness and compactness of the generalized composition operators on the space H ω using the ω-Carleson measures. Moreover, we give a lower bound for the essential norm of these operators.

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Abstract. Suppose ϕ is an analytic self-map of open unit disk D and w is an analytic function on D . Then a weighted composition operator induced by ϕ with weight w is given byWe find a sufficient condition under which two composition operators lie in the same path component of C (H 2 ) , and we find a sufficient condition for the difference of such operators to be compact on H 2 (D) . Then we provide another example that answers a question raised by Shapiro and Sundberg [18] negatively. Moreover, we characterize the Hilbert-Schmidt difference of two composition operators on H 2 (D) .Mathematics subject classification (2010): 47B38, 47B15, 47B33.