Journal of Operator Theory

Volume 43, Issue 1, Winter 2000 pp. 199-210.

On the commutant of the direct sum of operators of multiplication by the independent variable

Authors: B. Khani Robati (1), and K. Seddighi (2)
Author institution: (1) Department of Mathematics, College of Science, Shiraz University, Shiraz 71454, Iran
(2) Center for Theoretical Physics and Math., Department of Mathematics, P. O. Box 11365--8486, Tehran 11365, Iran

Summary: Let ${\cal B}$ be a direct sum of spaces of functions on each of which the operator $M_z$ of multiplication by $z$ $(f\rightarrow zf)$ is bounded. We determine the commutant of the direct sum of the operators of multiplication by $z$ on certain Hilbert spaces of functions (Banach spaces of functions). Also we characterize the commutant of $M_z$ and multipliers of Lipschitz algebras. Let $\mu$ be a compactly supported measure on ${\bbb C} $ and $t\ge 1$. We determine the commutant of the operator $M_z$ on $P^t(\mu)$, the closure of polynomials in $L^t(\mu)$, thus extending a result of M. Raphael for the case $t=2$.

Keywords: Commutant, muliplication by $z$, bounded point evaluation, Lipschitz algebra, direct sum of spaces

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