Subnormal weighted shifts and asymptotic properties of normal operators

Abstract

By Fuglede's theorem, an operator which commutes with a normal operator N must also commute with all spectral projections of N. We show that only in special cases is the theorem true when near-commutativity replaces commutativity. We show too that the restriction of a normal operator to an reducing subspace may be far from all normal operators on the subspace. Introduction. One of the formulations of the Fuglede theorem is that if N is a normal operator on a Hilbert space and A is an operator such that NA=AN, then E(Y)A=AE(59) where E is the unique spectral measure associated with N and ,Y is an arbitrary Borel subset of the plane. It is not unreasonable to conjecture an asymptotic generalization of this fact. That is, if N is a normal operator, S9 a Borel subset of the plane, and A an operator such that INA-ANJI is small, then IIAE(Y)-E(Y)AII is small. Precisely, for every positive number s, there exists a positive number 6 such that j1NA-ANII<6 implies JIE(Y)A-AE() 11 <e. In the same vein, it is very easy to show that if N is a normal operator and VX' is a subspace that reduces N, then N restricted to .Xf is also a normal operator. Again an asymptotic generalization of this fact may be conjectured as follows: If N is a normal operator and P is a projection onto a subspace X which almost reduces N, then the compression of N to XI is not far from the set of normal operators on *X; i.e., for every positive number E, there exists a positive number 6, such that 11 NP-PNJJ < 6 implies inf{lIPNPiran(,)-N'`J}<s, where the infimum is taken over all normal operators N' on S. In this paper it is shown that both conjectures are true in finite-dimensional Hilbert space, and both are false in infinite-dimensional Hilbert space. Received by the editors March 15, 1973. AMS (MOS) subject classifications (1970). Primary 47B15, 47B20; Secondary 47A20, 47B05.