On Laguerre-Sobolev matrix orthogonal polynomials

In this manuscript, we study some algebraic and differential properties of matrix orthogonal polynomials with respect to the Laguerre-Sobolev right sesquilinear form defined by < p , q > S & colone; integral 0 infinity p * ( x ) W L A ( x ) q ( x ) d x + M integral 0 infinity ( p ' ( x ) ) * W ( x ) q ' ( x ) d x , {\langle p,q\rangle }_{{\bf{S}}}:= \underset{0}{\overset{\infty }{\int }}{p}<^>{* }\left(x){{\bf{W}}}_{{\bf{L}}}<^>{{\bf{A}}}\left(x)q\left(x){\rm{d}}x+{\bf{M}}\underset{0}{\overset{\infty }{\int }}{(p<^>{\prime} \left(x))}<^>{* }{\bf{W}}\left(x)q<^>{\prime} \left(x){\rm{d}}x, where W L A ( x ) = e - lambda x x A {{\bf{W}}}_{{\bf{L}}}<^>{{\bf{A}}}\left(x)={e}<^>{-\lambda x}{x}<^>{{\bf{A}}} is the Laguerre matrix weight, W {\bf{W}} is some matrix weight, p p and q q are the matrix polynomials, M {\bf{M}} and A {\bf{A}} are the matrices such that M {\bf{M}} is non-singular and A {\bf{A}} satisfies a spectral condition, and lambda \lambda is a complex number with positive real part.