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Extra resources for A Banach Space Where Minimal Weak Cuscos are Generically Compact-Valued

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We mention also the earlier papers [36, 23] that inspired much of investigation of these and other classes of rational matrix-valued functions with symmetries. We now recall some of the arguments in [7], then explain the difficulties appearing in the several complex variables setting, and why the arguments of [7] extend to the non-commutative framework. So let F be a rational function which is J-unitary on the imaginary line, and assume that F is holomorphic in a neighborhood of the origin. It then admits a minimal realization F (z) = D + C(IIγ − zA)−1 zB where D = F (0), and A, B, C are matrices of appropriate sizes (the size γ × γ of the square matrix A is minimal possible for such a realization).

51 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3 More on observability, controllability, and minimality in the non-commutative setting . . . . . . . . . . . . . . . . . . . . 60 4 Matrix-J-unitary formal power series: A multivariable non-commutative analogue of the line case . . . . . . . 1 Minimal Givone–Roesser realizations and the Lyapunov equation . . . . . . . . . . .

85 6 Matrix-J-inner rational formal power series . . . . . . . . . . . . . . 1 A multivariable non-commutative analogue of the half-plane case . . . . . . . . . . . . . . . . . . . . . . . . 2 A multivariable non-commutative analogue of the disk case . . . . 91 7 Matrix-selfadjoint rational formal power series . . . . . . . . . . . . . 1 A multivariable non-commutative analogue of the line case . . . . 2 A multivariable non-commutative analogue of the circle case .

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