PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 97. Number 3. July 1986

THE GROUP GENERATEDBY UNIPOTENT OPERATORS1

C. K. FONG AND A. R. SOUROUR

Abstract. 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 canoni- cal 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

Received by the editors December 19, 1984 and, in revised form, April 23, 1985. 1980 Mathematics Subject Classification. Primary 47B99, 47D10; Secondary 15A30. 1This research has been supported by NSERC grants U0072 and A3674

¡B1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page 453

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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 L("f~), there exists a unipotent Usuch that UA is cyclic.

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 poly- nomial 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 C 0 1 0

0 0 -a «-i

We will show that there exist complex numbers tv t2,..., tn_l such that if ßi h o ... o 0 ß2 t2 ... o D '„-I 0 o o A,

then DC has eigenvalues ylt y2,..., y„. The proof is by induction on n. There is nothing to prove for n = 1. Now suppose the above is true for (« — 1) X (« — 1) matrices (n > 2), and let C be as above. Let g(x) = (x - y,) • • • (x - y„). We can write g(x) in the form g(0) + xgx(x) for a monic polynomial gj(x) of degree » — 1. Let rx be a root of the polynomial equation g,(x) — axß2 ••■/?„ = 0. It follows that /, satisfies the equation xgx(x) — tla1ß2 • • • ß„ = 0, and so there exists a monic polynomial h(x) of degree n - 1

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such that

xgi(x) - txarf2 ■■ ■ ß„ = (x - tx)h(x).

It follows that h(0) = alß2 ■■ ■ ßn, and so the product of the zeros of h(x) is (-l)""1a1j82 ••• ßn. By the induction hypothesis, there exist complex numbers t2,..., tn_l such that the characteristic polynomial of D0C0 is h(x), where C0 is the companion matrix of the polynomial ax + a2x + ■• • +an_lx" 1 + x"; and where D0 is the matrix with ß2,...,ßn on the main diagonal, t2,...,t _j on the diagonal above the main one, and zero everywhere else. Now let D be the n X n matrix with main diagonal ßx,...,ßn, with tx,...,t„_1 on the diagonal above, and with other entries zero. We have

0 ■■•0 m ßi DC = 0 AA 0

where ju = -(a0ßx + a^). By direct computation, we see that the characteristic polynomial of DC is

(x-tx)det(xI-D0C0)-uß2 ■■■ß„

= (x - tl)h(x) + a0Ußt + h^Ußi

= xgi(x) + aQßxß2 ■■■ ß„ = g(x),

so the eigenvalues of DC are yx,..., y„. This ends the induction proof. D The previous theorem implies that every cyclic A with determinant 1 is a product of two unipotents. Combining this with Lemmas 1 and 2, we have the following theorem.

Theorem 2. Every linear transformation with determinant 1 is a product of three unipotent transformations.

The infinite-dimensional case. We consider a separable infinite-dimensional Hil- bert space JA over the complex field. The algebra of all bounded linear operators on 3tif is denoted by B(Jif), and the group of all invertible operators is denoted by GL(Jf ). Let ^ be the set of all unipotent operators of order 2. The set of all products of n operators in <2fis denoted by 01". For an operator T, an equation such as T = ((■ BD)means that there is a unitary operator from JC to Jf ffi 3^ which carries T to the operator (¡i BD)on Jf © Jf. We start with two lemmas.

Lemma 3. If both A and 1 + A are invertible, then A © A l e <%2.

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Proof. Let 'lAil+AY1 (A - 1)(1 + A)~lS FÁA) (1-^)(1+^)_1 2(1 +A)-1 'lAil+A)'1 (A-1- 1)(1 +AY1 F2(A) (A2-A)(l+AYl 2(1+^)_1 Then both FX(A), F2(A) are in °U, and A © A~x = Fl(A)F2(A). D Lemma 4. If \a\ = 1 and C is invertible, then al e C e *4. Proof. Choose e > 0 such that e < 1 and e||C|| < 1. Write D = eC. Now al © C is unitarily equivalent to • • • © al © al C al © al © which is the product of the following two operators:

A = ■■ ■ ffia4D © ä2D~l © a2D © Dl © \d~\ © ae_1l © äd © aV1!

B = ■■■ ©ä3£>-1 © a3D © äD1 © aD e_1l © el aV1! a2el © ■ Following the proof of Lemma 3, we have a"D © a~"D-1 = F1(a"D)F2(a"D), â"él © «"e^l = /1(ä''d)F2(ä',d). Notice that sup^F^D^l < oo, sup„||F2(a"D)\\ < oo, sup„||JF1(ä"el)|| < oo and sup„||F2(ä"£l)|| < oo. Thus A is unitarily equivalent to the product AXA2, where Aj. = ■• • ®Fj(a4D) © F;(a2Z)) © Fj(D) © ^(äd) © Fj(â3â) © • • • (y = 1,2). It follows from the proof of Lemma 1 that both Ax and A2 are in <%,and hence A e <%2.In the same way we have B e ^2. Therefore al © C g aUA. D Theorem 3. GL( Jf ) = ^6. Proof. We consider two cases. In the first case, we assume that T is not a scalar modulo the compacts. By a theorem of Brown and Pearcy [1, Corollary 3.4], T is similar to an operator of the form (* f). Notice'Hi that Ï):

where A is some invertible operator. (That A is invertible follows from the fact that T is invertible.) Also, a Te ïhî ?)■ where A*= -OT1. By Lemma 4, (¡J °) e W4. Since T is similar to 1 B)l 1 Q\(A 0^ ,0 i A-a- 1A0 1 we have re«6.

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It now remains to consider the case T = XI + K, where K is compact. We notice that X + 0, since T is invertible. Let e be a positive real number to be determined later. By Riesz theory, K is similar to K0 © F, where K0 has spectral radius < e, and where F is an operator on a finite-dimensional space. By a result of Rota [6, Theorem 2], K0 is similar to an operator whose norm is < e, so we may assume without loss of generality that \\K0\\ < e. Write

¡Kl K2

and let A = (X + Kx + K2)~\a - X - Kx), where a = 1 if X # ± 1 and a = / if X = +1. (The operator X + Kx + K2 is invertible if e is small enough.) Let e /1 + A -A ) S-{ A I-Al We have (S — l)2 = 1. Furthermore. a X — a + Lx (X + K0)S= *. a -X + Li 2X - a + L3 where Lx= Kx + K2, L2 = (K3 - Kx)(l + A) +(KA - K2)A and LZ = KX + K4 +(KX + K2- K3- K4)A. Let 1 0 1 5X - 1 - ÔX, 1

Therefore

i; *\j where J = (1 - Xa)(L2 - Lx) + L3 - clLxL2. We notice that by our choice of a we have a # äX2. From the expressions for /, Lx, L2, L3 we see that J is compact, and that we can choose e small enough so that ||7|| < \a - aX2\; soaí a(aX2 + J). Now we have

R(X + K0)S=[aQ *) with« <£o(C).

It follows by [5, Corollary 0.15] that R(X + K0)S is similar to al © C. Since (R © 1)[(X + K0) ©(X + F)](S © 1) = R(X + K0)S ffi(X + F), and since T is similar to (X + A:0) © (X + F), we have (R © 1)T(S © 1), similar to al © C © (X + F), which, by Lemma 4, belongs to <%4.Since R © 1 and S © 1 are unipotent, we get ie*6. This ends the proof of the theorem. D

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REFERENCES 1. A. Brown and C. Pearcy, Structure of commutators of operators, Ann. of Math. (2) 82 (1965), 112-127. 2. K. Hoffman and R. Kunze, Linear algebra, 2nd ed., Prentice-Hall, Englewood Cliffs, N.J., 1971. 3. I. Kaplansky, Fields and rings, University of Chicago Press, Chicago, 111.,1972. 4. H. Radjavi, The group generated by involutions, Proc. Roy. Irish Acad. Sect. A 81 (1981), 9-12. 5. H. Radjavi and P. Rosenthal, Invariant subspaces. Springer-Verlag, Berlin, 1973. 6. G.-C. Rota, On models for linear operators, Comm. Pure Appl. Math. 13 (1960), 469-472. Department of Mathematics, University of Toronto, Toronto, Ontario M5S 1A1, Canada Department of Mathematics, University of Victoria, Victoria, British Columbia V8W 2Y2, Canada (Current address A. R. Sourour)

Current address (C. K. Fong): Department of Mathematics, University of Ottawa, Ottawa, Ontario KIN 9B4, Canada

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