An Extension of the Weyl-von Neumann Theorem to Normal Operators^)

We prove that a normal operator on a separable Hubert space can be written as a diagonal operator plus a compact operator. If, in addition, the spectrum lies in a rectifiable curve we show that the compact operator can be made HilbertSchmidt. In 1909 Hermann Weyl proved [3] that each bounded Hermitian operator on a separable Hubert space can be written as the sum of a diagonal operator and a compact operator. J. von Neumann proved in 1935 [2] that the compact operator can actually be made Hilbert-Schmidt and also observed that boundedness is unnecessary. P. R. Halmos has inquired in [1] whether the Weyl result, and, in his 1969 Brazilian lectures, whether the von Neumann result can be extended to normal operators. We prove here that Weyl's result can be so extended. That is, each normal operator can be written as the sum of a diagonal operator and a compact operator. If, in addition, the spectrum lies in a rectifiable curve, von Neumann's result holds. That is, the compact operator can actually be made Hilbert-Schmidt. In each of these cases the bounded case is the crucial one; the extension to unbounded operators is easy. It is also easy to see that, as in the Weyl and von Neumann results, the norm of the compact operator or the Hilbert-Schmidt norm of the Hilbert-Schmidt operator respectively may be made arbitrarily small. We conjecture that, in general, for normal operators, the compact operator cannot be made Hilbert-Schmidt. Indeed, we conjecture that the operator given by multiplication by z on L2[7x7] cannot be so decomposed; the positive 2dimensional Lebesgue measure of the purely continuous spectrum of this operator seems to prevent the compact operator from being Hilbert-Schmidt. We were first made acquainted with these questions by a lecture of P. R. Halmos at the regional conference sponsored by the NSF at Texas Christian University in May 1970. We gratefully acknowledge valuable discussion with J. Dyer, P. Halmos and H. Porta. We use standard terminology. However, for economy of notation we will allow ourselves the looseness of speaking of an unbounded operator "on" 77. We use Received by the editors October 8, 1970. AMS 1970 subject classifications. Primary 47B15, 47B05, 47B10.