Szegö's limit theorem: The higher-dimensional matrix case

The continuous version of Szegö's theorem gives the first two terms of the asymptotics as α → ∞ of the determinants of certain convolution operators on L₂(0, α) with scalar-valued kernels. Generalizations are known if the kernel is matrix valued or if the interval (0, α) is replaced by αΩ with Ω a bounded set in Rⁿ with smooth boundary. This paper treats the higher-dimensional matrix case. The coefficient in the interesting (second) term is an integral over the contangent bundle of ?Ω of the correponding coefficients of one-dimensional problems.