The tensor phase under a tensor–tensor product

The concept phase has been shown to play an important role in the analysis of multi-input multi-output system. Motivated by the recent results representing the multilinear time-invariant system in terms of standard notions of tensors, the definition of phases for third-order complex sectorial tensors via the tensor–tensor product (T-product) is proposed as a natural counterpart in this paper. The properties of tensor phases are shown along with sectorial tensor decompositions, compressions, Schur complements, tensor–tensor T-product, tensor sum, Hadamard product and T-Kronecker products. Furthermore, the phases of banded sectorial tensor completion and banded sectorial tensor decomposition are studied. Besides, we derive a tensor version of the Kalman–Yakubovich–Popov lemma.