# IJPAM: Volume 54, No. 3 (2009)

INTEGRAL REPRESENTATIONS OF UNBOUNDED
OPERATORS BY INFINITELY SMOOTH
BI-CARLEMAN KERNELS

Igor M. Novitskii
Institute for Applied Mathematics
Far-Eastern Branch of the Russian Academy of Sciences
Dzerzhinskiy Street 54, Khabarovsk, 680 000, RUSSIA
e-mail: novim@iam.khv.ru

Abstract.In this paper, we establish that if a closed linear operator in a separable Hilbert space is unitarily equivalent to a bi-Carleman integral operator in an appropriate , then that operator is unitarily equivalent to a bi-Carleman integral operator in , whose kernel and two Carleman functions , are infinitely smooth and vanish at infinity together with all partial and all strong derivatives, respectively. The implementing unitary operator (from onto ) is found by direct construction.