Abstract

In this paper, we investigate densely defined Toeplitz composition operators TuCφ on the Fock space F2. We completely characterize the Jλ-complex symmetry of such operators and derive necessary and sufficient conditions for normality. We further establish full criteria for self-adjointness and construct an explicit example showing that an operator may be Jλ-complex symmetric without being normal. To extend the theory beyond fixed radial–trigonometric expansions, we combine the Gaussian-weighted Mellin transform with Adaptive Fourier Decomposition (AFD) and establish the unified AFD-Mellin Symmetry Criterion, which generalizes these symmetry characterizations to adaptive rational basis systems.

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