IJPAM: Volume 51, No. 3 (2009)

OPERATORS WITH SLOWLY GROWING RESOLVENTS
TOWARDS THE SPECTRUM

Paul O. Oleche$^1$, N. Omolo-Ongati$^2$, John O. Agure$^3$
$^{1,2,3}$Department of Mathematics
Maseno University
P.O. Box 333, Maseno, KENYA
$^1$e-mail: [email protected]
$^2$e-mail: [email protected]
$^3$e-mail: [email protected]


Abstract.A closed densely defined operator $H$, on a Banach space X, whose spectrum is contained in R and satisfies

\begin{displaymath}
\norm{(z-H)^{-1}}\; \leq \; c
\frac{\Abs{z}^{\alpha}}{\abs{\Im
z}^{\beta}} \qquad \forall \; z \; \not \in \fld{R}
\end{displaymath} (1)

for some $\alpha \;,\; \beta \geq 0$; $c > 0$, is said to be of . If instead of ([*]) we have
\begin{displaymath}
\norm{(z-H)^{-1}}\; \leq \; c
\frac{\abs{z}^{\alpha}}{\abs{\Im
z}^{\beta}} \qquad \forall \; z \; \not \in \fld{R},
\end{displaymath} (2)

then $H$ is of .

Examples of such operators include self-adjoint operators, Laplacian on $L^1(\fld{R})$, Schrödinger operators on $L^p(\fld{R}^n)$ and operators $H$ whose spectra lie in R and permit some control on e^iHt.

In this paper we will characterise the $\context{\type{\alpha}{\beta}}$ operators. In particular we show that property ([*]) is stable under dialation by real numbers in the interval (0,1) and perturbation by positive reals. We will also show that is $H$ is of then so is $H^2$.

Received: March 13, 2008

AMS Subject Classification: 47A10

Key Words and Phrases: spectrum, resolvent, eigenvalues, diagonalizable, scale invariant

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 51
Issue: 3