Bounded Semigroups of Matrices

In this note are proved two conjectures of Daubechies and Lagarias. The first asserts that if Z is a bounded set of matrices such that all left infinite products converge, then 8 generates a bounded semigroup. The second asserts the equality of two differently defined joint spectral radii for a bounded set of matrices. One definition involves the conventional spectral radius, and one involves the operator norm. The occurrence of convergent infinite products of matrices pervades many current areas of mathematics. See, for example, the various articles in [6]. Recently, in studying curve and surface generation, several authors [1,2,5] have been led to sets of matrices all (or almost all) infinite products of which converge. Although the contexts vary, this infinite product convergence seems to be a fundamental underlying phenomenon. Thus in [5] Micchelli and Prautzsch are motivated by subdivision methods, in [2] Daubechies and Lagarias are motivated by wavelets and dilation equations, and in [l] Berger is motivated by iterated function systems and random algorithms for curve and surface generation. In [2] Daubechies and Lagarias explore sets of matrices all infinite products of which converge. This note presents some additional results in that direction, and studies the general structure of bounded semigroups of *Currently visiting the Department of Applied Mathematics and Computer Science, Weizmann Institute of Science, Rehovot 76100, Israel. Research supported by Air Force Ofice of Scientific Research, Grant AFOSR-87-0137. LZNEAR ALCEBRA AND ITS APPLZCATZONS 16621-27 (1992) 0 Elsevier Science Publishing Co., Inc., 1992 21 655 Avenue of the Americas, New York, NY 10010 0024-3795/92/$5.00 22 MARC A. BERGER AND YANG WANG matrices. It also proves two conjectures made in [2], called by them the boundedness conjecture and the generalized spectral radius conjecture. Let .k = An denote the algebra of all real rr X n matrices, and let 11. ]I be a norm on R(“). This norm induces a corresponding operator norm 11. II on A?. Let Z c A? be a nonempty bounded set of matrices, and denote by 4 = S(Z) the semigroup generated by Z augmented with the n X n identity matrixZ=Z,,~othat4=U”,~,~“,where~~=~im,~M,:M~~~,l~ii m}. We say that Z is LCP (left convergent products) if every infinite product from I: left converges, i.e., if lim, em M, . . . M, exists for any sequence (Mi)y= 1 in 2,. In this case denote by 2” the set of all such limits. Define llzll= sup{ IlMll: M E 2)) m l/m p^=pI(‘c) = limsup]]x I] . m-rm The quantity b(z) is a special case of the joint spectral radius of a bounded subset of a normed algebra defined in [7]. Observe that p^ does not depend on the particular choice for the norm on .A THEOREM I (a) Product boundedness. Zf C is LCP then 4 is bounded. (That is, z is product bounded, using the terminology from [2].) In particular p^ < 1. (b) s is LCPwith x”=O ifandonZyiff<<. Proof. (a): Let X be the subspace X= 1 x ELF!(“): sup ]]Sx]l Then X is invariant under each M E C, and by the uniform boundedness principle I( 41x I] < 03. Suppose X is not all of lR(“‘. CLAIM. Vx G X, C > 0 there is S E 9 such that SxtEX and llSxll>C. Indeed, since x 4 X, there exists S’= M;** M, with Mie15, l max(l, llJl~II*ll2ll)C. (1) If S’x G X then simply take S = S’. Otherwise choose k < m such that M, . . -M,xEX but Mk+l-.-Mlx~X.