Convergence of matrix continued fractions

Generalized continued fractions: a unified definition and a Pringsheim-type convergence criterion

“…yields the recursion X n = X n-1 b n + X n-2 a n . Such continued fractions in non-commutative structures have been introduced in [12,13,33,45,46], further convergence results can be found in [1,19,25,37,39,41,47].…”

“…They assumed b k , c k , a k , d k to be matrices with dimensions independent of k, θ k being a quadratic matrix. Here, we assume that all coefficients are elements of some Banach algebra R. In order to guard against misunderstandings (in the literature, the term 'matrix continued fractions' is also used for continued fractions with matrix-valued coefficients, see [33,37,41,47]), we will prefer referring to Levrie and Bultheel's construction as LB-fractions.…”

“…This criterion is referred to as Worpitzky-type criterion, and in fact, it is older than Śleszyński-Pringsheim-type criteria. In the literature, -for non-generalized continued fractions in Banach algebras defined by X n = X n-1 b n + X n-2 a n (that is, c n+1,n = I), Śleszyński-Pringsheim-type and/or Worpitzky-type criteria can be found in [1,13,19,25,37,39,47], -for non-generalized two-sided continued fractions in Banach algebras, such criteria were proven in [2,11,39], -for (finite) complex-valued Jacobi chains, a Śleszyński-Pringsheim-type criterion can be found in [28], -for complex-valued n-fractions as defined by de Bruin, Śleszyński-Pringsheim-type criteria are proved in [21,22]. Proving a Pringsheim-type criterion for gcfs includes all of these results.…”

Continued Fraction Expansion of the Heinz Operator Mean

“…The real case is relatively well studied in the literature ( [9][10]). However, in contrast to the theoretical importance, one can find in mathematical literature only a few results on the continued fractions with matrix arguments ( [12][13][14][15][16]).…”

Application of matrix-valued integral continued fractions to spectral problems on periodic graphs with defects

“…As a conclusion note that there are only few papers devoted to applications of (finite or infinite) standard matrix-valued continued fractions (MCF) to spectral problems: in [6] some general relations between Hamiltonians and MCF are presented; in [7] MCF are applied for calculating Green functions related to some Hamiltonians; in [8] the authors use MCF in analysis of non-linear spectral problems; in [9] some methods of obtaining Floquet eigenvalues and eigensolutions based on MCF are discussed; in [10], [11] the stability of the methods is analyzed. Some applications of MCF to explicit representations of resolvent operators can be found in [12], [13].…”