Higher Degree $t$-Hermitian Forms and Positivity-Preserving Contractions

Abstract

In this article we introduce higher-degree $t$-Hermitian forms, a tubal analogue of ordinary Hermitian forms of arbitrary degree. Through a synthesis of multilinear matrix multiplication and the $t$-product on third-order tensors, we show that $t$-Hermitian forms are in bijection with odd order tubal tensors satisfying certain symmetry conditions, which we call $t$-conjugate partial symmetry. After applying the Fast Fourier Transform along the tubal mode of the corresponding tubal tensor, $t$-Hermitian forms decompose into a family of classical Hermitian forms. This decomposition enables us to characterize positivity of $t$-Hermitian forms in terms of the spectra of the conjugate partially symmetric Fourier slices of its corresponding tubal tensor, yielding a tubal analogue of the spectral theorem for classical higher degree Hermitian forms. Then, we study classical Hermitian forms induced by contractions along the tubal mode of $t$-conjugate partially symmetric tensors, characterizing when such contractions preserve positivity and deriving quantitative lower bounds for the positivity margin of the resulting classical Hermitian forms.