$H$-Operator Approximation Spaces via Delayed Riesz Means and Quasi-Banach Moduli

We establish a constructive and quasi-Banach framework for approximation spaces $A_{\mu }^{\rho }$ of compact $H$-operators between Banach and quasi-Banach spaces. By leveraging delayed Riesz means $V_{2^{bn}}$ generated by self-adjoint differential operators $P(D)$, we replace abstract best approximants with explicit linear operator decompositions. Furthermore, we extend the theory of $H$-operator approximation spaces to quasi-Banach settings ($0<p<1$), bypassing the collapse of Peetre’s $K$-functional by employing localized moduli of smoothness ${\omega }_{\phi }^r(f,t{)}_p$. Integrating seminal spectral bounds due to Markus \cite{M1966} and factorization techniques for operator ideals, we prove that operators belonging to $A_{\mu }^{\rho }$ possess complete systems of root vectors whose expansions are Abel summable.