arXiv:2106.03266v1 [math.FA] 6 Jun 2021 Weighted Composition Operator and the Mittag-Leffler space HIMANSHU SINGH Abstract. We present important characterizations of the Weighted Composition Operator over the Mittag Leffler space of entire functions. These characterizations include the Hilbert-Schmidt and Unitary char- acterizations of the Weighted Composition Operator over the Mittag- Leffler space. We also see the result for the spectrum of Weighted com- position operator over the Mittag-Leffler space. Keywords. Mittag Leffler space, Littlewood-Paley Identity, Fock space, Reproducing Kernel Hilbert space, Compact, Weighted Composition Operator, Caputo fractional derivative. 1. Introduction Considering holomorphic mappings ψ and φ over C, the Weighted Compo- sition Operator W ψ,φ over the Mittag-Leffler space ML 2 (C; q) induced by ψ : C → C and φ : C → C is formally defined as follows: W ψ,φ f = ψ · f ◦ φ, where f ∈ ML 2 (C; q). These operators arises naturally. The fundamental root of the weighted composition operator was laid by Forelli in 1964, where he showed that, an isometry on H p for p = 2 and 1 <p< ∞ is actually a weighted composition operator in [For64]. These operators have been very well studied on various kinds of Hilbert spaces. That being stated, Jury studied the weighted com- position operator on the Hardy space H 2 in [Jur07]. On the other hand, we have results for weighted composition operator on Fock space by Le [Le14]. A thorough investigation on the operator theoretic aspects of W ψ,φ was recently done by Hai and Rosenfeld [HR21], where boundedness and com- pactness of W ψ,φ were at the center stage of the paper. Recently, Singh gave new formulations for the compactness of W ψ,φ over ML 2 (C; q) by devising the Littlewood-Paley Identity for ML 2 (C; q) in [Sin21a]. For more details on the Littlewood-Paley Identity on different Hilbert spaces of holomorphic