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\n Denote by\n \n \n \n \n \n W<\/mml:mi>\n <\/mml:mrow>\n 1<\/mml:mn>\n <\/mml:msub>\n \\mathcal {W}_1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n the set of complex valued functions of the form\n \n \n \n \n a<\/mml:mi>\n (<\/mml:mo>\n z<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n \n \n \u2211\n \n <\/mml:mo>\n \n i<\/mml:mi>\n =<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n \n +<\/mml:mo>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:munderover>\n \n a<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n \n z<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n a(z)=\\sum _{i=-\\infty }^{+\\infty }a_iz^i<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n such that\n \n \n \n \n \n \n \u2211\n \n <\/mml:mo>\n \n i<\/mml:mi>\n =<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n \n +<\/mml:mo>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:munderover>\n \n |<\/mml:mo>\n <\/mml:mrow>\n i<\/mml:mi>\n \n a<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n \n |<\/mml:mo>\n <\/mml:mrow>\n ><\/mml:mo>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n \\sum _{i=-\\infty }^{+\\infty }|ia_i|>\\infty<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We call QT-matrix a quasi-Toeplitz matrix\n \n \n \n A<\/mml:mi>\n A<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , associated with a symbol\n \n \n \n \n a<\/mml:mi>\n (<\/mml:mo>\n z<\/mml:mi>\n )<\/mml:mo>\n \n \u2208\n \n <\/mml:mo>\n \n \n W<\/mml:mi>\n <\/mml:mrow>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n a(z)\\in \\mathcal W_1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , of the form\n \n \n \n \n A<\/mml:mi>\n =<\/mml:mo>\n T<\/mml:mi>\n (<\/mml:mo>\n a<\/mml:mi>\n )<\/mml:mo>\n +<\/mml:mo>\n E<\/mml:mi>\n <\/mml:mrow>\n A=T(a)+E<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n \n T<\/mml:mi>\n (<\/mml:mo>\n a<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n (<\/mml:mo>\n \n t<\/mml:mi>\n \n i<\/mml:mi>\n ,<\/mml:mo>\n j<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n )<\/mml:mo>\n \n i<\/mml:mi>\n ,<\/mml:mo>\n j<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n +<\/mml:mo>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n T(a)=(t_{i,j})_{i,j\\in \\mathbb {Z}^+}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the semi-infinite Toeplitz matrix such that\n \n \n \n \n \n t<\/mml:mi>\n \n i<\/mml:mi>\n ,<\/mml:mo>\n j<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n =<\/mml:mo>\n \n a<\/mml:mi>\n \n j<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n i<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n t_{i,j}=a_{j-i}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , for\n \n \n \n \n i<\/mml:mi>\n ,<\/mml:mo>\n j<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n +<\/mml:mo>\n <\/mml:msup>\n <\/mml:mrow>\n i,j\\in \\mathbb Z^+<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and\n \n \n \n \n E<\/mml:mi>\n =<\/mml:mo>\n (<\/mml:mo>\n \n e<\/mml:mi>\n \n i<\/mml:mi>\n ,<\/mml:mo>\n j<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n )<\/mml:mo>\n \n i<\/mml:mi>\n ,<\/mml:mo>\n j<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n +<\/mml:mo>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n E=(e_{i,j})_{i,j\\in \\mathbb {Z}^+}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a semi-infinite matrix such that\n \n \n \n \n \n \n \u2211\n \n <\/mml:mo>\n \n i<\/mml:mi>\n ,<\/mml:mo>\n j<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n \n +<\/mml:mo>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:munderover>\n \n |<\/mml:mo>\n <\/mml:mrow>\n \n e<\/mml:mi>\n \n i<\/mml:mi>\n ,<\/mml:mo>\n j<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n |<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n \\sum _{i,j=1}^{+\\infty }|e_{i,j}|<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n 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\n \n \n \n \n A<\/mml:mi>\n \n X<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n +<\/mml:mo>\n B<\/mml:mi>\n X<\/mml:mi>\n +<\/mml:mo>\n C<\/mml:mi>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n AX^2+BX+C=0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , encountered in the solution of Quasi-Birth and Death (QBD) stochastic processes with a denumerable set of phases, is presented where\n \n \n \n \n A<\/mml:mi>\n ,<\/mml:mo>\n B<\/mml:mi>\n ,<\/mml:mo>\n C<\/mml:mi>\n <\/mml:mrow>\n A,B,C<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n are QT-matrices.\n <\/p>","DOI":"10.1090\/mcom\/3301","type":"journal-article","created":{"date-parts":[[2017,6,29]],"date-time":"2017-06-29T10:47:55Z","timestamp":1498733275000},"page":"2811-2830","source":"Crossref","is-referenced-by-count":21,"title":["Semi-infinite quasi-Toeplitz matrices with applications to QBD stochastic processes"],"prefix":"10.1090","volume":"87","author":[{"given":"Dario","family":"Bini","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stefano","family":"Massei","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Beatrice","family":"Meini","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2018,1,24]]},"reference":[{"key":"1","unstructured":"B. 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