IJPAM: Volume 56, No. 2 (2009)

SIMULTANEOUS INTEGRAL REPRESENTABILITY BY
INFINITELY SMOOTH KERNELS WITH APPLICATION
TO INTEGRAL EQUATIONS

Igor M. Novitskii
Khabarovsk Division
Institute for Applied Mathematics
Far-Eastern Branch of the Russian Academy of Sciences
54, Dzerzhinskiy Street, Khabarovsk, 680 000, RUSSIA
e-mail: [email protected]


Abstract.In this note, we characterize families incorporating those bounded linear operators on a separable Hilbert space $\mathcal{H}$ that can be simultaneously transformed by the same unitary equivalence transformation into Carleman integral operators on $L^2(\mathbb{R})$, whose kernels $\boldsymbol{T}\colon\mathbb{R}^2\to\mathbb{C}$ and Carleman functions $\overline{\boldsymbol{T}(s,\cdot)}\colon\mathbb{R}\to L^2(\mathbb{R})$ are infinitely smooth and vanish at infinity together with all partial and all strong derivatives, respectively. An explicit procedure for constructing the unitary operators, from $\mathcal{H}$ onto $L^2(\mathbb{R})$, effecting such transformations is also presented. As an application, we present a smooth version of Korotkov's reduction method for general third-kind integral equations in $L^2(Y,\mu)$, whose aim is to obtain an equivalent integral equation of either the first or the second kind in $L^2(\mathbb{R})$, with an infinitely smooth Hilbert-Schmidt or Carleman kernel, respectively.

Received: September 7, 2009

AMS Subject Classification: 47G10, 45P05, 47B33, 47B38, 47N20

Key Words and Phrases: bounded Hilbert space operator, Hilbert-Schmidt operator, bounded integral linear operator, Carleman integral operator, characterization theorems for integral operators, linear integral equation of the third kind

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 56
Issue: 2