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

RESUMO

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 $\omega_\varphi^r(f,t)_p$. 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.