proceedings of the american mathematical society Volume 12, Number 3. March 1994 VON NEUMANN'S INEQUALITY FOR COMMUTING, DIAGONALIZABLE CONTRACTIONS. II B. A. LOTTO AND T. STEGER (Communicated by Theodore W. Gamelin) Abstract. We construct a triple T = {Tx, Ti, T$) of commuting, diagonalizable contractions on C5 a polynomial p in three variables for which p(r) > HpIIoo, where p oo denotes the supremum norm of p over the unit polydisk in C3. 1. Introduction In part I [2], the first author showed that von Neumann's inequality (1) \\P(T)\\ < IIPlU holds for all polynomials p in n variables, where 7 is an n-tuple of commuting, diagonalizable contractions on C^ that satisfies some additional hypotheses. Here p(7) denotes the operator norm of p(7) \\p\\oc denotes the supremum norm of p over the unit polydisk of C". In the present work we present an example to show that the extra hypotheses cannot be removed. Our example is based on an example due to Kaijser Varopoulos [3, addendum] that shows that (1) can fail with n = 3 N = 5. This example consists of nilpotents; our example is obtained by perturbing their example to diagonalizables. 2. The counterexample Theorem. There are three commuting, diagonalizable contractions Tx, T2, Ti on C5 a polynomial p in three variables such that \\p(tx, T2, Ti)\\ > llplloo - We start with the example of Kaijser Varopoulos mentioned above. Use the stard inner product on C5. The three operators / 1 \ Ax=, \ l/y/1 -l/vi -1/V3 / Received by the editors November 1, 1991, in revised form, July 1, 1992; presented at AMS meeting #876, Dayton, Ohio, October 1992. 1991 Mathematics Subject Classification. Primary 47A63; Secondary 15A6. 897 1994 American Mathematical Society
898 B. A. LOTTO AND T. STEGER ( \ A2= 1, \ -l/v7! l/v7! -1/V3 / / \ ^3= 1 \ -1/^/3 -l/>/3 1//3 / on C5 are commuting contractions, the polynomial p(zx, z2, z3) = z\ + z\ + z\- 2zxz2-2zxzs - 2z2z3 satisfies \\p\\oc = 5 \\p(ax, ^2, A{)\\ > 5. We will produce, for every e >, perturbations A(p of Aj (j = 1,2,3) that commute are diagonalizable, with the additional property that A{p -* Aj as e -». Since then ^e) -> \\Aj\\ = 1 as e -», we may replace Af> by ^e)/im*e) assume that A(J] is a contraction. We will then have \\p(a^,a1t),af)\\^\\p(ax,a2,ai)\\>5 as e ->, so that for small enough e we have \\p(a\e), A2e), A{^)\\ > 5. Setting Tj = A^-p for such an e gives the theorem. We therefore need only construct the perturbations. Let ( 1\ 1-1-1 X= -1 1-1, -1-1 1 \1 / let Y be any matrix such that Y~x = YlX, that is, such that YYl = X~x. We can produce one such Y as follows. Since X~x is real symmetric, there is a real orthogonal matrix U a diagonal matrix D such that X-1 = UDU1. Let -/D be a diagonal matrix whose square is D set Y = Uy/DU1. By replacing Y by TO, where O is a suitably chosen orthogonal matrix, we may assume that the fifth row of Y contains only nonzero entries. We denote the fifth row by y think of it as an element of C5. Consider the linear map L from C5 into operators on C5 defined by La = YA Yl, where A is the diagonal matrix whose diagonal is a. The fifth row of La is (y * a)yl, where * denotes coordinatewise multiplication. Since Y is invertible y has only nonzero entries, it follows that the linear map that sends a to the fifth row of La is invertible. Hence, we can find ux, u2, «3 in C5 such that the fifth rows of Lux, Lu2, L«3 are (, 1,,,),
VON NEUMANN'S INEQUALITY. II 899 (,, 1,,), (,,, 1,), respectively. Since La is always symmetric, we have For e > / = 1,2,3, diagonal is Uj. We have /* * * * \ * * * * 1 Lux = * * * * * * * *, \ 1 / /* * * * \ * * * * Lu2 = * * * * 1, * * * * \ 1 / /* * * * \ * * * * Lut, = * * * *. * * * * 1 \ 1 / let /-e) denote the diagonal matrix whose ^ * * * *N YUl )Y~x = YU[i)YtX = e(lui)x= * * * *, * * * * \ e -e -e / where the missing entries are constant multiples of e. Let De be the diagonal matrix with diagonal entries (1, e~x, e_1, e~x, e~2/y/3). Then / * * * *\ (2) (D(Y)Uf](Y-xD-x)= * * * *, * * * * \ 1/^/3-1/V3-1/V3 / where the missing entries are constant multiples of e, e2, or e3. This is the perturbation A^. Letting Ze = D(Y, we similarly set /fj * * * *\ * * * * (3) Af = ZtUfZ-x = 1 * * * * * * * * \ -1/v^ 1A/3 -l/v7? /
9 B. A. LOTTO AND T. STEGER / * * * *\ * * * * (4) A(p = Z(U^]Z-X = * * * *. VP -\/y/l -l/\/3 I/n/3 / Clearly, A^, A2 ), A^ commute are diagonalizable. The missing entries of (3) (4), like those of (2), are constant multiples of e, e2, e3, so Af - Aj as e -+ for j = 1, 2, 3. We have found our perturbations the theorem is proved. 3. Questions The above proof involves perturbing a triple of commuting matrices to commuting, diagonalizable ones. Question 1. Can any triple of commuting operators on a finite-dimensional space be perturbed to become commuting diagonalizable? It is known that any pair of commuting matrices can be perturbed to commuting diagonalizables [1]. The natural generalization to four matrices fails, as the following example shows. Let {ei,..., e } be the stard basis for C". We define n by n matrices that act as follows: Tiej = ej+i for 1 < j < n - 3, 72e _i = en, 73ei = e, 74e _i = e -2, where Tkej = in all other cases. The product of two distinct Tk is zero, so these matrices commute. The algebra with identity generated by the Tk*s has (T()"Zq, 72, 73, 74 as a basis so has dimension n + 1. If we could perturb the T^'s to commuting diagonalizables, the algebra generated by the perturbations would be an (n + 1 )-dimensional commutative algebra of diagonalizable n by n matrices. But no such algebra exists. This argument by dimension suggests the following questions. Question 2. Consider a subalgebra of the n by n matrices, commutative with identity. If this subalgebra has dimension no greater than n, can it be perturbed to a commutative subalgebra of diagonal matrices? Question 3. Can a triple of commuting n by n matrices generate an algebra with identity of dimension greater than n? Question 2 also suggests stabilizing an arbitrary collection of commuting n by n matrices. Question 4. Given a finite collection of commuting matrices, can their direct sums with a large enough zero matrix be perturbed to commuting diagonalizables?
von neumann's inequality. ii 91 References 1. J. A. R. Holbrook, Polynomials in a matrix its commutant, Linear Algebra Appl. 48 (1982), 293-31. 2. B. A. Lotto, Von Neumann s inequality for commuting, diagonalizable contractions. I, Proc. Amer. Math. Soc. 12 (1994), 889-895. 3. N. Th. Varopoulos, On an inequality of von Neumann an application of the metric theory of tensor products to operator theory, J. Funct. Anal. 16 (1974), 83-1. Department of Mathematics, University of California, Berkeley, California 9472 E-mail address: lottotaath.berkeley. edu Current address: Department of Mathematics, Vassar College, Poughkeepsie, New York, 1261 E-mail address: BeLottoQvassar. edu Department of Mathematics, University of Chicago, Chicago, Illinois 6637 E-mail address: stegerqzaphod.uchicago. edu Current address: Department of Mathematics, University of Georgia, Boyd Graduate Studies, Athens, Georgia 362-743 E-mail address: stegerq j oe. math. uga. edu