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 $\mathcal{H}$ is unitarily equivalent to a bi-Carleman integral operator in an appropriate $L^2(Y,\mu)$, then that operator is unitarily equivalent to a bi-Carleman integral operator in $L^2(\mathbb{R})$, whose kernel $\boldsymbol{T}:\mathbb{R}^2\to\mathbb{C}$ and two Carleman functions $\boldsymbol{t}(s)=\overline{\boldsymbol{T}(s,\cdot)}$, $\boldsymbol{t}^{\boldsymbol{\prime}}(s) =\boldsymbol{T}(\cdot,s):\mathbb{R}\to L^2(\mathbb{R})$ are infinitely smooth and vanish at infinity together with all partial and all strong derivatives, respectively. The implementing unitary operator (from $\mathcal{H}$ onto $L^2(\mathbb{R})$) is found by direct construction.

Received: June 6, 2009

AMS Subject Classification: 47G10, 45P05, 47B33, 47B38

Key Words and Phrases: closed linear operator, integral linear operator, Carleman integral operator, bi-Carleman integral operator, characterization theorems for integral operators, linear integral equation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 54
Issue: 3