The motivation of the paper is a close investigation of inverse spectral problems and the proof of few uniqueness theorems for isospectral sets of periodic matrix-valued Schrödinger operators. That is to determine spectral data (interpreted in a very broad sense) which uniquely determine the matrix-valued coefficients in Schrödinger, Jacobi and Dirac type systems. It begins with a brief review of the basic Weyl-Titchmarsh theory for matrix-valued Schrödinger- and Dirac-type operators as needed in the sequel.
The principal result proven is that, for a fixed \(x_0\in\mathbb{R}\) and all \(z\in\mathbb{C}_+\), \(g(z,x_0)\) and \(g'(z,x_0)\) uniquely determine the matrix-valued \(m\times m\) potential \(Q(z)\) for a.e. \(x\in\mathbb{R}\), where \(g(z,x)\) denotes the diagonal Green matrix of the selfadjoint \(m\times m\)-matrix-valued Schrödinger operator \[ H= -{d^2\over dx^2} I_m+ Q\quad\text{in }L^2(\mathbb{R})^m,\quad m\in\mathbb{N}. \] A local version of this result is also given as an adjoint. The last section is devoted to the investigation of the Weyl-Titchmarsh theory and uniqueness theorems for matrix-valued Jacobi (i.e., second-order finite differences) operators.
The paper not only presents the material as stated above, it also mentions in a synchronous manner almost all the previous work related to the topic discussed. This is evident from the fact that it is supplemented with 145 references.