Normal-equivalent operators and operators with dual of scalar-type

If T ∈ L(X) is such that T′ is a scalar-type prespectral operator, then Re T′ and Im T′ are both dual operators. It is shown that that the possession of a functional calculus for the continuous functions on the spectrum of T is equivalent to T′ being scalar-type prespectral of class X, thus answering a question of Berkson and Gillespie.

Products of generalized equivalent operators in angular momentum theory

Generalized operator equivalents to the (bipolar) spherical harmonics [Formula: see text] are considered, viz. [Formula: see text], where the PJ are projection operators on the manifolds of definite angular momenta, A closed formula is derived for the coefficients of the Clebsch–Gordan decomposition of products of such operators.