On the Diagonal of an Operator1 By-

Characterizations of zero-diagonal operators (i.e., operators that have a diagonal whose entries consist entirely of zeros) and the norm-closure of these operators are obtained. Also included are new characterizations of trace class operators, self-commutators of bounded operators, and others. 0. Introduction. In [3], it is shown that a hermitian compact operator is a self-commutator of a compact operator if and only if it has a zero-diagonal. Motivated by this result, we decided to launch an investigation into the connection between the diagonal of an operator and the operator itself; we wonder to what extent an operator can be described by its diagonal. It turns out that, as expected, the diagonal of an operator carries more information about the operator than its relatively small size (compare to the "fat" matrix representation of the operator) may suggest. Let H be a (complex, separable) Hilbert space, and B(H) the space of all (bounded) operators on H. By zero-diagonal operators we mean the operators 7 such that there exists an orthonormal basis {bj} for H satisfying (Tbf, bf) = 0 for all j. It is not hard to see that the set of these operators is pathwise connected (in fact, all its elements can be connected to the origin 0 inside itself); that it is not a linear manifold (21 ® ( — 1) and (-1) ® 21 belong to it but their sum I ® I does not); and moreover, that it stays away from /. (Indeed, the distance is no less than one: Take a zero-diagonal operator 7 and choose an orthonormal basis {bj} so that (Tbj, bf) = 0. Then ||7 /|| = sup|W|.,((7 I)x, x) > ((7 /)*/,, ft,) = 1.) In the first section we are to characterize zero-diagonal operators and hermitian zero-diagonal operators. §2 is devoted to the identification of the norm-closure of zero-diagonal operators. It turns out that this is a quite well-known class of operators. As a result, we are able to obtain new characterizations for some related classes of operators—among them, self-commutators of bounded operators and trace class operators. 1. Zero-diagonal operators. Let H be a finite-dimensional Hilbert space and 7 an operator acting on H. Then the following three statements are equivalent: (1) 7 has a zero-diagonal; Received by the editors May 19, 1981 and, in revised form, May 3, 1983. 1980 Mathematics Subject Classification. Primary 47A12, 47B99; Secondary 47B10, 47B47.