Abstract: 

For an unbounded operator $S$ the question
whether its subnormality can be built up from that of every $S_f$,
the restriction of $S$ to a cyclic space generated by $f$ in the
domain of $S$, is analyzed. Though the question at large has
been left {\it open}\/ some partial results are presented and a
possible way to prove it is suggested as well.

1. Instytut Matematyki
   Uniwersytet Jagielloński
   Reymonta 4
   30-059 Kraków
   Poland