Author’s abstract: Let \((X, h)\) be a compact and irreducible Hermitian complex space of complex dimension \(m\). In this paper we are interested in the Dolbeault operator acting on the space of \(L^2\) sections of the canonical bundle of \(\mathrm{reg}(X)\), the regular part of \(X\). More precisely let \(\overline{\mathfrak{d}}_{m, 0} : L^2 \Omega^{m, 0}(\mathrm{reg}(X), h) \rightarrow L^2 \Omega^{m, 1}(\mathrm{reg}(X), h)\) be an arbitrarily fixed closed extension of \(\overline{\partial}_{m, 0} : L^2 \Omega^{m, 0}(\mathrm{reg}(X), h) \rightarrow L^2 \Omega^{m, 1}(\mathrm{reg}(X), h)\) where the domain of the latter operator is \(\Omega_c^{m, 0}(\mathrm{reg}(X))\). We establish various properties such as closed range of \(\overline{\mathfrak{d}}_{m, 0}\), compactness of the inclusion \(\mathcal{D}(\overline{\mathfrak{d}}_{m, 0}) \hookrightarrow L^2 \Omega^{m, 0}(\mathrm{reg}(X), h)\) where \(\mathcal{D}(\overline{\mathfrak{d}}_{m, 0})\), the domain of \(\overline{\mathfrak{d}}_{m, 0}\), is endowed with the corresponding graph norm, and discreteness of the spectrum of the associated Hodge-Kodaira Laplacian \(\overline{\mathfrak{d}}_{m, 0}^\ast \circ \overline{\mathfrak{d}}_{m, 0}\) with an estimate for the growth of its eigenvalues. Several corollaries such as trace class property for the heat operator associated to \(\overline{\mathfrak{d}}_{m, 0}^\ast \circ \overline{\mathfrak{d}}_{m, 0}\), with an estimate for its trace, are derived. Finally in the last part we provide several applications to the Hodge-Kodaira Laplacian in the setting of both compact irreducible Hermitian complex spaces with isolated singularities and complex projective surfaces.