Semi-infinite quasi-Toeplitz matrices with applications to QBD stochastic processes

Source abstract

Denote by W 1 \mathcal {W}_1 the set of complex valued functions of the form a ( z ) = ∑ i = − ∞ + ∞ a i z i a(z)=\sum _{i=-\infty }^{+\infty }a_iz^i such that ∑ i = − ∞ + ∞ | i a i | > ∞ \sum _{i=-\infty }^{+\infty }|ia_i|>\infty . We call QT-matrix a quasi-Toeplitz matrix A A , associated with a symbol a ( z ) ∈ W 1 a(z)\in \mathcal W_1 , of the form A = T ( a ) + E A=T(a)+E , where T ( a ) = ( t i , j ) i , j ∈ Z + T(a)=(t_{i,j})_{i,j\in \mathbb {Z}^+} is the semi-infinite Toeplitz matrix such that t i , j = a j − i t_{i,j}=a_{j-i} , for i , j ∈ Z + i,j\in \mathbb Z^+ , and E = ( e i , j ) i , j ∈ Z + E=(e_{i,j})_{i,j\in \mathbb {Z}^+} is a semi-infinite matrix such that ∑ i , j = 1 + ∞ | e i , j | \sum _{i,j=1}^{+\infty }|e_{i,j}| is finite. We prove that the class of QT-matrices is a Banach algebra with a suitable sub-multiplicative matrix norm. We introduce a finite representation of QT-matrices together with algorithms which implement elementary matrix operations. An application to solving quadratic matrix equations of the kind A X 2 + B X + C = 0 AX^2+BX+C=0 , encountered in the solution of Quasi-Birth and Death (QBD) stochastic processes with a denumerable set of phases, is presented where A , B , C A,B,C are QT-matrices.