On the rank of a spectral function

M. S. Brodskii

Matematičeskie zametki, Tome 5 (1969) no. 4, pp. 457-460.

Résumé

Let $P(x)$, $0\leqslant x\leqslant1$, be an absolutely continuous spectral function in the separable Hilbert spaces $\mathfrak{S}$. If the vectors $h_j$, $j=1,2,\dots,s$, $s\leqslant\infty$ are such that the set $P(x)h_j$ is complete in $\mathfrak{S}$, then the rank of the function $P(x)$ equals the general rank of the matrix-function $d/dx||P(x)h_i,h_j||^s_1$.