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⟨Z,W ⟩ = Z1W ∗ 1 + · · ·+ ZnW ∗ n , where W ∗ j is a conjugate transpose of the matrix Wj . The set B m,n = {(Z1, . . . , Zn) = Z ∈ C [m×m] : I − ⟨Z,Z⟩ > 0} , is called a matrix ball (of the first type); here ⟨Z,Z⟩ = Z1Z 1 + Z2Z 2 + · · · + ZnZ n is the ‘scalar’ product, I is the identity [m ×m]-matrix, Z∗ ν = Z ′v is the conjugate transpose of Zν , ν = 1, 2, . . . , n, [5]. Here I −⟨Z,Z⟩ > 0 means that the Hermitian matrix I −⟨Z,Z⟩ is positively defined, i.e. all its eigenvalues are positive.

Boundary Morera Theorem for the Matrix Ball of the Third Type