THE FUGLEDE COMMUTATrvTTY THEOREM MODULO OPERATOR IDEALS

Let H denote a separable, infinite-dimensional complex Hilbert space. A two-sided ideal / of operators on H possesses the generalized Fuglede property (GFP) if, for every normal operator N and every X e L(H), NX XN e I implies N*X XN* e /. Fuglede's Theorem says that / = {0} has the GFP. It is known that the class of compact operators and the class of Hilbert-Schmidt operators have the GFP. We prove that the class of finite rank operators and the Schatten /»-classes for 0 < p < 1 fail to have the GFP. The operator we use as an example in the case of the Schatten /»-classes is multiplication by z + w on L2 of the torus. Introduction. Let 77 denote a separable, infinite-dimensional complex Hilbert space. Let 7.(77) d ä"(77) d Cp D F(H) (0 </» < oo) denote, respectively, the class of all bounded linear operators, the class of compact operators, the Schatten /»-class, and the class of finite rank operators on 77. All operators herein are assumed to be linear and bounded. Let || • \\p denote the C^-norm. Let 7 be any two-sided ideal in 7.(77) (every ideal herein is assumed to be two-sided). It is well known that if 7 i* {0} or 7.(77), then AT(77) D 7 D F(77). Definition. The ideal 7 is said to possess the generalized Fuglede property (GFP) if, for every normal operator N and every bounded operator X, we have NX — XN E I implies A'*AXN* E I (i.e., NX = XN modulo 7 implies N*X = XN* modulo 7). Fuglede's Theorem essentially states that 7 = {0} has the GFP [1]. There is a connection between the GFP for Cx (which is not known to hold true) and a possible generalization of the trace result [5, Question 3]. Namely, if A is a normal operator and A" is a compact operator such that NX — XN E Cx, must trace(AA" XN) = 0? If C, possessed the GFP, then the answer to this question would be yes. The proof is the same as the proof for the well-known fact that if S is a selfadjoint operator and X is a compact operator then SA' — XS E C, implies trace(SX XS) = 0 (cf. [8, p. 279, Lemma 1.3]). Say T = NX XN E C,. If C, possessed the GFP, then S = N*X XN* E Cx. Hence -S* = NX* X*N G C, and F S* = N(X + X*) (X + X*)N and F + S* = N(X A"*) (X — X*)N. But X + X* and X — X* are scalar multiples of compact, selfadjoint Received by the editors February 13, 1980 and, in revised form, May 19, 1980 and August 27, 1980. 1980 Mathematics Subject Classification. Primary 47A30, 47B05, 47B10, 47B15, 47B47; Secondary 47A55, 47A05, 47D25.