On the Square Roots of Infinite Matrices

Abstract

Is there a matrix A which has the property A2 = C? We began by constructing lower triangular matrices which have this property. Alerted by a colleague to work of Hausdorff, Toeplitz, and others, we investigated various convergence properties of our square roots. We used computer-generated estimates of the first 25 rows of two of the square roots to formulate conjectures which we then proved on the behavior of their entries. We showed that the first of these square roots takes convergent sequences to convergent sequences and that the second does not even take bounded sequences to bounded sequences. For a while we believed that all the square roots of C were lower triangular; eventually, having found no obstruction, we proved that all nonzero infinite matrices have nontriangular square roots. Also, our technique for finding lower triangular square roots of C applies to any infinite lower triangular matrix with all positive entries on the diagonal. Halmos was, of course, interested in finding a square root of C which is also a bounded operator, i.e., one which takes square-convergent series to square-convergent series. J. B. Conway has proved the existence of a well-defined bounded subnormal operator vC by using the Conway-Olin functional calculus [1], [2]. Also, by a theorem of T. Kato [3], C has a bounded accretive square root. The proofs are not constructive, so we do not know a matrix square root which is a bounded subnormal square root. In our investigation, we used only elementary techniques. The computations are complex, but require knowledge only of calculus, linear algebra, binomial coefficients and, above all, the principle of induction. Most of the proofs are inductive; the square roots are iteratively defined; the inductive relationship between binomial coefficients reflected in Pascal's triangle is essential. In this note we present our results without proof, but will gladly supply details upon request.