The Group Generated by Unipotent Operators1

The group generated by unipotent n X n complex matrices is SL„(C), and every member of the latter is a product of three unipotent matrices. The group generated by unipotent operators on Hilbert space Jf is GL(Jf ), and every invertible operator is a product of six unipotent operators of order 2. An operator U on a Hilbert space is called unipotent [3, p. 100] if U = 1 + N, where N is nilpotent. It is called unipotent of order n if N" = 0 and N"-1 # 0. Our aim is to characterize the group S generated by the unipotent operators and to show that every element of S is a product of a small number of unipotents. In a finite-dimensional space it is obvious that every operator in S has determinant one. We prove the converse of this. More precisely, we show that every operator with determinant 1 is a product of three unipotents. In the infinite-dimensional case we show that 'S is the group of all invertible operators and that every operator in 'S is a product of six unipotents of order 2. We point out that other generators for the group of invertible operators were obtained by Radjavi [4]. The finite-dimensional case. In what follows "V is an «-dimensional vector space over the complex field, where n is finite. The algebra of all linear transformations on y is denoted by L{i^). A linear transformation T on "V is called cyclic if there exists a vector x such that the vectors x, Tx, T2x,. ..,Tnlx span "f. It is well known that T is cyclic if and only if its Jordan canonical form contains only one Jordan block for every eigenvalue, and that this is the case if and only if the minimal polynomial of T equals its characteristic polynomial (see [2, Chapter 7]). As usual, the group of « X « complex matrices with determinant 1 is denoted by SL„(C). Lemma 1. For each invertible A e L(f") with a singleton spectrum, there exists a unipotent Usuch that UA is cyclic and o(UA) = a(A). Proof. Let o(A) = {X}, and let N = A XI. By considering the Jordan canonical form, we see that there exists a basis {eve2,...,en} of ~V such that Ne¡ = ejej + 1 (1 <_/'<« — 1) and Nen = 0, where e/ = 0 or 1. Let S be the usual shift; i.e., Sej = e/+1 (1 </ < n 1) and Se„ = 0, and then let J = S N. If U = 1 + A"1/, Received by the editors December 19, 1984 and, in revised form, April 23, 1985. 1980 Mathematics Subject Classification. Primary 47B99, 47D10; Secondary 15A30. 1 This research has been supported by NSERC grants U0072 and A3674 ¡B1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page 453 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 454 C. K. FONG AND A. R. SOUROUR then U is unipotent and UA = \l + S S2D for some transformation D which is diagonal with respect to the basis under consideration. Using the fact that DS = SS*DS and using induction, we get that for every positive integer m, {UA Xl)m = Sm + Sm+lRm for some Rm s L(*"). In particular, (UA XI)""1 * 0, but (UA XI)" = 0. This shows that o(UA) = {X} and that UA is cyclic. D Lemma 2. For each invertible A cyclic. L("f~), there exists a unipotent Usuch that UA is Proof. We may write A as a direct sum A = A1 ® ■ ■ ■ ®Ak, where o(Aj) = {Xj} and X, # Xj for / =£ j. By Lemma 1, there are unipotents U, such that U-Aj is cyclic and a(UjAj)= (Xy) for every j. If U = Ul ffi ■ • • © £/fc, then UA is cyclic, since it is the direct sum of cyclic operators with disjoint spectra. D The following theorem may be of independent interest. Theorem 1. Let A be a cyclic linear transformation, and let /?, and y, (1 < í < m) be complex numbers such that de\.AY\"=xßt■ = n"_i y¡. Then there exists a linear transformation B such that B has eigenvalues ßl,...,ßn and BA has eigenvalues Yi, • • •, Y„. The eigenvalues are repeated according to algebraic multiplicity. Proof. Let f(x) = aQ + axx + ■ ■ ■ +an_lxn'1 + x" be the characteristic polynomial of A. By taking an appropriate basis, we may assume that the matrix of A is the companion matrix of f(x)—i.e., the matrix 0 0 0 1 0 0 0 1 0 C 0 0 -a «-i We will show that there exist complex numbers tv t2,..., tn_l such that if