Semi-normal operators

Abstract

As in § 1.5, a bounded operator T will be called semi-normal if

$$T{{T}^{*}}-{{T}^{*}}T=D,\ D\geqq 0\ \operatorname{or}\ D\leqq 0.$$

(3.1.1)

Clearly any normal operator is also semi-normal. It is easy to see that the converse is also true in case the space H is finite-dimensional. For if, say, D≧0, its eigenvalues are non-negative while their sum is the trace of D, which is 0. Hence all eigenvalues are 0 and so D=0. In the infinite-dimensional case however it is possible that an operator be semi-normal without being normal. In fact, any isometric but not unitary operator V has this property; for V* V= I and VV* − V*V ≦ 0, ≠0. On l² the operator A given by the matrix A= (a _ij ) with a _{i +1},i=1 and a _ij = 0 otherwise (i, j =1, 2, ...) is such an operator.