Let (X, h) be a compact and irreducible Hermitian complex space of complex dimension m. In this talk we will report some recent results concerning the Hodge-Kodaira Laplacian acting on the space of L^2 sections of the canonical bundle of reg(X), the regular part of X. More precisely, given any L^2 closed extension D of the Dolbeault operator acting on the space of smooth (m, 0)-forms with compact support in reg(X), we will consider the operator D*D, which is a self-adjoint extension of the Hodge-Kodaira Laplacian. We will show that D*D is a discrete operator, that the heat operator associated to D*D is trace class and moreover we will provide an estimate for its trace and for the growth of the eigenvalues of D*D. Finally we will apply these results in the setting of complex projective surfaces. Here we will show that the absolute extension of the Hodge-Kodaira Laplacian in bi-degree (2, q), with q=0, ..., 2 , is a discrete operator. Moreover we will show that the corresponding heat operators are trace class, providing an estimate for their traces and for the growth of the eigenvalues.