Operator-Inequality Defects in Transform-Based Tensor Algebras: Invariance and Transform Sensitivity

Transform-based tensor algebras induced by invertible linear transforms provide a unified framework for tensor operations and spectral analysis. This paper investigates how the choice of a unitary transform affects the quantitative tightness of classical operator inequalities. We first formulate a transform-domain transfer principle for matrix trace inequalities and examine its consequences for the Golden–Thompson, Jensen-type, Klein, and Lieb inequalities. We then study the Golden–Thompson defect through its exact slice-wise decomposition. Under bounded spectral assumptions, we establish an upper estimate in terms of squared transform-domain commutators and a matching-order lower estimate for an explicit Pauli-matrix family. We further establish qualitative transform sensitivity: explicit separation results for real orthogonal transforms, including a DFT–DCT separation at p=4. Finally, we investigate the non-universality of transform selection and formulate commutator-energy minimization as an operator-aware surrogate objective over the orthogonal group. The framework separates algebraic invariance from transform-dependent inequality tightness without assuming general equivalence between the defect and commutator-energy objectives.