Inverses of infinite sign regular matrices

de Boor, C.; Friedland, S.; Pinkus, A.

doi:10.1090/S0002-9947-1982-0670918-7

Inverses of infinite sign regular matrices
HTML articles powered by AMS MathViewer

by C. de Boor, S. Friedland and A. Pinkus PDF

Trans. Amer. Math. Soc. 274 (1982), 59-68 Request permission

Abstract:

Let $A$ be an infinite sign regular (sr) matrix which can be viewed as a bounded linear operator from ${l_\infty }$ to itself. It is proved here that if the range of $A$ contains the sequence $( \ldots ,1, - 1,1, - 1, \ldots )$, then $A$ is onto. If ${A^{ - 1}}$ exists, then $D{A^{ - 1}}D$ is also sr, where $D$ is the diagonal matrix with diagonal entries alternately $1$ and $- 1$. In case $A$ is totally positive (tp), then $D{A^{ - 1}}D$ is also tp under additional assumptions on $A$.

References

Carl de Boor, The inverse of a totally positive bi-infinite band matrix, Trans. Amer. Math. Soc. 274 (1982), no. 1, 45–58. MR 670917, DOI 10.1090/S0002-9947-1982-0670917-5
A. S. Cavaretta Jr., W. Dahmen, C. A. Micchelli, and P. W. Smith, On the solvability of certain systems of linear difference equations, SIAM J. Math. Anal. 12 (1981), no. 6, 833–841. MR 635236, DOI 10.1137/0512069
Ronald A. DeVore and Karl Scherer (eds.), Quantitative approximation, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 588164
Samuel Karlin, Total positivity. Vol. I, Stanford University Press, Stanford, Calif., 1968. MR 0230102
Charles A. Micchelli, Infinite spline interpolation, Approximation in Theorie und Praxis (Proc. Sympos., Siegen, 1979) Bibliographisches Inst., Mannheim, 1979, pp. 209–238. MR 567661