Geodesic
On the rank of a spectral function
Matematičeskie zametki, Tome 5 (1969) no. 4, pp. 457-460

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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 $d/dx||P(x)h_i,h_j||^s_1$. 
@article{MZM_1969_5_4_a8,
     author = {M. S. Brodskii},
     title = {On the rank of a spectral function},
     journal = {Matemati\v{c}eskie zametki},
     pages = {457--460},
     publisher = {mathdoc},
     volume = {5},
     number = {4},
     year = {1969},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1969_5_4_a8/}
}
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M. S. Brodskii. On the rank of a spectral function. Matematičeskie zametki, Tome 5 (1969) no. 4, pp. 457-460. http://geodesic.mathdoc.fr/item/MZM_1969_5_4_a8/