Matrix Szegő Function and Matrix Orthogonal Polynomials for Multiple Cuts

Following recent developments on matrix-valued orthogonal polynomials (MVOPs), we construct the matrix Szeg\H{o} factorisation of the matrix weight function when it is supported on multiple intervals. We find that on the gaps the matrix function has unitary multiplicative jumps. In the next part, performing the Deift-Zhou steepest descent analysis we study the associated global parametrix which comes from the Riemann-Hilbert problem of the MVOPs. A solution is constructed visualising the rows of the parametrix as sections of vector bundles on a hyper-elliptic Riemann surface. An alternate view point to the global parametrix is also presented using a vanishing lemma. As an application we obtain strong asymptotics of MVOPs for multiple cuts which extends the works of Dea\~no, Kuijlaars, and Rom\'an.