The integers \(\nu_1\geq\nu_2\geq\dots\), \(\lambda_1\geq\lambda_2\geq\dots\), and \(\mu_1\geq\mu_2\geq\dots\) that are the sizes of the Jordan cells of a nilpotent matrix \(T\) and its restrictions \(T| M\), \(T^*| M^{\perp}\) (with \(M\) an invariant subspace of \(T\)), respectively, were characterized in terms of the Littlewood–Richardson rule by T. Klein [J. Lond.Math.Soc.43, 280–284 (1968; Zbl 0188.09504)]. Two continuous versions of the Littlewood–Richardson rule were proposed in [H. Bercovici, W. S.Li and T. Smotzer, Adv.Math.134, No. 2, 278–293 (1998; Zbl 0919.47007); W. S.Li, V. Müller, Acta Sci.Math.(Szeged) 64, No. 3–4, 609–625 (1998; Zbl 0921.47012)].
In the paper under review, it is shown that these two versions are equivalent and both provide a necessary and sufficient condition for the Jordan model of a \(C_0\) contraction \(T\), \(T| M\), \(T^*| M\). These rules are also used to find the nonnegative eigenvalues of compact selfadjoint operators. The authors prove necessary and sufficient conditions for sequences that appear as nonnegative eigenvalues of compact selfadjoint operators \(A,B,C\) satisfying \(A+B=C\). This generalizes a result of A. Klyachko [Sel.Math., New Ser.4, No. 3, 419–445 (1998; Zbl 0915.14010)] concerning Hermitian matrices.