Advanced Search

Let ${\beta \left(n\right)}_{n=0}^{\infty }$ be a sequence of positive numbers and 1 ≤ p < ∞. We consider the space ${\ell }^p\left(\beta \right)$ of all power series $f\left(z\right)={\sum }_{n=0}^{\infty }f̂\left(n\right)zⁿ$ such that ${\sum }_{n=0}^{\infty }{\left|f̂\left(n\right)\right|}^p{\left|\beta \left(n\right)\right|}^p<\infty$. We give some sufficient conditions for the multiplication operator, $M_z$, to be unicellular on the Banach space ${\ell }^p\left(\beta \right)$. This generalizes the main results obtained by Lu Fang [1].

We consider Hilbert spaces of analytic functions on a plane domain Ω and multiplication operators on such spaces induced by functions from $H^{\infty }\left(\Omega \right)$. Recently, K. Zhu has given conditions under which the adjoints of multiplication operators on Hilbert spaces of analytic functions belong to the Cowen-Douglas classes. In this paper, we provide some sufficient conditions which give the converse of the main result obtained by K. Zhu. We also characterize the commutant of certain multiplication operators.

Let ${\left\{\beta \left(n\right)\right\}}_{n=0}^{\mathrm{\infty }}$ be a sequence of positive numbers and $1\le p<\mathrm{\infty }$. We consider the space $H^p\left(\beta \right)$ of all power series $f\left(z\right)={\sum }_{n=0}^{\mathrm{\infty }}\stackrel{⌃}{f}\left(n\right)z^n$ such that ${\sum }_{n=0}^{\mathrm{\infty }}{\left|\stackrel{⌃}{f}\left(n\right)\right|}^p\beta {\left(n\right)}^p<\mathrm{\infty }$ . Suppose that $\frac{1}{p}+\frac{1}{q}=1$ and ${\sum }_{n=1}^{\mathrm{\infty }}\frac{n^{qj}}{\beta {\left(n\right)}^q}=\mathrm{\infty }$ for some nonnegative integer $j$. We show that if $C_{\phi }$ is compact on $H^p\left(\beta \right)$, then the non-tangential limit of ${\phi }^{\left(j+1\right)}$ has modulus greater than one at each boundary point of the open unit disc. Also we show that if $C_{\phi }$ is Fredholm on $H_p\left(\beta \right)$, then $\phi$ must be an automorphism of the open unit disc.