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Invertibly convergent infinite products of matrices, with applications to difference equations. (English) Zbl 0846.40002

Definition 1. If \(\{B_j\}\) are \(k\times k\) matrices, we define \[ \prod^s_{m=r} B_j= \begin{cases} B_s B_{s-1} \dots B_r &\text{if } r\leq s,\\ I &\text{if } r>s, \end{cases} \] thus, successive terms multiply on the left.
Definition 2. An infinite product \(\prod^\infty_{m=1} B_m\) of \(k\times k\) matrices converges invertibly if there is an integer \(N\) such that \(B_m\) is invertible for \(m\geq N\) and \[ Q= \lim_{n\to \infty} \prod^n_{m=N} B_m \] exists and is invertible.
In this paper the author has proved the following theorems:
Theorem 1. The infinite product \(\prod^\infty_{m=1} (I+ A_m)\) converges invertibly if and only if, for some integer \(N\geq 1\), the difference equation \(X_{n+1}= (I+ A_n) X_n\), \(n\geq N\), has a solution \(\{X_n \}^\infty_{n=N}\) such that \(\lim_{n\to \infty} X_n =I\). In this case \[ P_n= X_{n+1} X_N^{-1} \prod^{N- 1}_{m= 1} (I+ A_m) \qquad \text{and} \qquad P= X_N^{-1} \prod^{N-1}_{m=1} (I+ A_m). \] Theorem 2. Suppose that \(\sum^\infty A_m\) converges, \[ G_n= \sum^\infty_{m =n} A_m= O(\rho_n) \qquad \text{and} \qquad \sum^\infty |G_{m+1} A_m |< \infty. \] Let \(H_n= \sum^\infty_{m=n} G_{m+1} A_m= O(\sigma_n)\), and define \(\Phi_n= \max (\rho_n, \sigma_n)\).
Then \(P= \prod^\infty_{m=1} (I+A_m)\) converges invertibly, and \(P_n= P+ O(\Phi_{n+1})\).
Theorem 3. Let \(\{B_m \}^\infty_{m=1}\) be a sequence of \(k\times k\) matrices and \(\{b_m\}^\infty_{m=1}\) a sequence of non-negative numbers such that \[ \biggl|\sum^n_{j=1} B_j\biggr|\leq b_n, \quad n \geq 1, \qquad \text{and} \qquad \limsup_{m\to \infty} |B_m |< \infty. \] Let \(\{C_m \}^\infty_{m=1}\) be a sequence of \(k\times k\) matrices such that \(\lim_{n\to \infty} b_m |C_m|=0\) and \( \sum^\infty b_m |C_m- C_{m+1} |< \infty\). Define \(\nu_n= b_{n-1} |C_n|+ \sum^\infty_{m=n} b_m |C_m- C_{m+1} |\), where \(b_0 =0\). Suppose that \(\sum^\infty \nu_{m+1} |C_m|<\infty\). Then the infinite products \[ P= \prod^\infty_{m=1} (I+ B_m C_m) \qquad \text{and} \qquad \widehat {P}= \prod^\infty_{m=1} (I+ C_m B_m) \] converge invertibly, and \[ \prod^n_{m=1} (I+ B_m C_m)= P+ O(\Phi_{n+1})\qquad \text{and} \qquad \prod^n_{m=1} (I+ C_m B_m)= \widehat {P}+ O (\Phi_{n+1}), \] where \[ \Phi_n= \max \Biggl( \nu_n, \sum^\infty_{m=n} \nu_{m+1} |C_m|\Biggr). \] Theorem 1 indicates the connection between invertible convergence of an infinite product of square matrices and the asymptotic properties of solutions of a linear system of difference equations. Theorem 2 is an extension of a result of A. Trgo [Monodromy matrix for linear difference operators with almost constant coefficients, J. Math. Anal. Appl. 194, No. 3, 697-719 (1995)]. Theorem 3 is a new result.
In this paper a standard definition of convergence of an infinite product of scalars is extended to the infinite product of square matrices.
Reviewer: S.Sharma (Ujjain)

MSC:

40A20 Convergence and divergence of infinite products
39A10 Additive difference equations
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References:

[1] A. Trgo, Monodromy matrix for linear difference operators with almost constant coefficients, J. Math. Anal. Appl.; A. Trgo, Monodromy matrix for linear difference operators with almost constant coefficients, J. Math. Anal. Appl. · Zbl 0843.39014
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[3] Wintner, A., On a theorem of Bôcher in the theory of ordinary linear differential equations, Amer. J. Math., 76, 183-190 (1954) · Zbl 0055.08104
[4] I’a Bromwich, T. J., An Introduction to the Theory of Infinite Series (1959), Macmillan and Company: Macmillan and Company New York · JFM 39.0306.02
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