Spectral-Conic Operator Space Theory: Geometric Foundations, Dual Structures, and Unified Spectral Optimization Framework

This paper constructs a novel mathematical structure—the spectral cone operator space. This structure extends vector convex cone theory to the eigenvalue geometry level and establishes spectral cone mappings, spectral cone operator spaces, spectral cone polar structures, and their dual systems. We prove the equivalence conditions for the closure and convexity of spectral cone sets, establish the spectral cone polar cone representation theorem, give the extremum characterization of the spectral cone norm, and prove the strong duality theorem for spectral cone programming under generalized conditions. This theory unifies linear cone programming, semidefinite programming, and spectral norm optimization at the structural level and provides a new operator geometry framework.