Diminishing of the spectrum of an operator under extension

A negative answer is given to the second question posed on page 210 of the book Complex Analysis — 199 Research Problems, Lecture Notes in Math., No. 1043, Springer, Berlin (1984).Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 45, pp. 96–97, 1986.