Multiplier Operators and Invariant Subspace Structures in Complex-Valued Harmonic Function Spaces

In this paper, we characterize subspaces of complex-valued harmonic functions in the unit disk in terms of prescribed dilatations, study their pointwise multiplier algebras, and investigate multipliers and invariant subspaces on the harmonic Hardy space. Motivated by the fact that the pointwise product of harmonic functions is, in general, not harmonic, we develop an operator-theoretic framework based on Hadamard (coefficientwise) convolution. We show that only the analytic and co-analytic dilatation classes admit nontrivial multiplier algebras under pointwise multiplication, while Hadamard convolution preserves the harmonic decomposition and induces a diagonal operator structure on the associated Hilbert space, allowing a complete characterization of convolution multipliers and their invariant subspaces. These results provide new insights into the interplay between harmonic function theory and operator theory in complex analysis.