IJPAM: Volume 30, No. 3 (2006)

THE NON-EXISTENCE OF RADIALLY ACTING
FREDHOLM INTEGRAL OPERATORS ON THE ANALYTIC
FUNCTION SPACE $\bf {A_2(D)}$ OF THE OPEN UNIT DISC $\bf {D}$

Attila Béla Von Keviczky$^1$, Domenic Elio Manzo$^2$
$^1$Department of Mathematics and Statistics
Concordia University
1455 de Maisonneuve Blvd. W., Montreal, QC, H3G 1M8, CANADA
e-mail: attila@mathstat.concordia.ca
$^2$7501 Rue Dolier
St. Leonard, QC, H1S 2J7, CANADA
e-mail: dominic.manzo@sympatico.ca


Abstract.It is herein demonstrated that radially acting Fredholm integral operators $(Kf)(re^{i\theta})=\int\limits_0^1K(r,r^\prime,\theta)f(r^\prime e^{i\theta})e^{i\theta}dr^\prime$ on the analytic function space $A_2(D)$, $K(r,r^\prime,\theta)$ having uniformly bounded double norm $\vert\vert\vert K(\cdot,\cdot,\theta)\vert\vert\vert$ on the unit square of $\mathbb R^2$ in the sense of $L_2[0,1]$, can only be given by a $\theta$-parameter family of $L_2$-Volterra kernels $K(r,r^\prime,\theta)$ - i.e. $K(r,r^\prime,\theta)=0\ ( 0 \leq r^\prime \leq r \leq 1,\ -\pi \leq \theta \leq \pi )$ - and hence determine a quasi-nilpotent operator $K\in {\mathcal L}(A_2(D))$. Representation in terms of the sesquilinear tensor product $A_{s2}(D)\oslash L_2[0,1]$ - i.e. $K(r,r^{\prime}, \theta)= \sum\limits_{n=0}^{\infty}(\cdot ^n\oslash k_n)(r,r^\prime,\theta)$ - is shown with $\vert\vert\vert\ \ \vert\vert\vert _s$-convergence if and only if $K(r,r^\prime,\theta)$ is uniformly square integrable on the triangle $\Delta \equiv \{ (r,r^\prime ):0\leq r^\prime\leq r\leq 1\}$ or equivalently $(\theta \mapsto K(\cdot,\cdot,\theta)) \in C([-\pi,\pi], L_2(\Delta))$. Uniform square integrability in terms of ``when do boundary values of $H(D)$-functions belong to the Banach algebra $\Lambda_{\alpha}$ or $\Lambda_*$" is discussed. Radial Fredholm and Hammerstein integral equations are solved in $A_{s2}(D)$ and $A_2(D)$. Finally, the holomorphic extension from $[0,1]$ to all of $\overline D$ of the solutions of ordinary linear differential equations, whose coefficients are restrictions of $H(\overline D)$-functions to the interval $[0,1]$, is explicitly given.

Received: June 10, 2006

AMS Subject Classification: 47E05, 46L07, 46L06, 46E20, 46E15, 45P05, 45G10, 45D05, 45B05, 34B27, 34B05, 28A15

Key Words and Phrases: radially acting integral operators of Fredholm, Volterra and Hammerstein type, tensor products, $\theta$-parameter family of $L_2$-kernels, convolutions in $L_2(-\pi,\pi)$, Dirichlet-kernels, weak-convergence, theorems of Cassorati-Weierstraß, Dini, Müntz-Szász and Zygmund, $\theta$-parameter family of Borel measures absolutely continuous with respect to Lebesgue measure, uniformly square-integrable kernels, t-constriction kernels, Banach algebras, Fredholm resolvents, spaces $\Lambda_\alpha$ and $\Lambda_*$, closed left ideals, Green's function, holomorphic extensions of solutions of boundary-value problems

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 30
Issue: 3