Weighted composition operators from Banach spaces of analytic functions into Bloch-type spaces

Let $X$ be a Banach space of analytic functions on the unit disk $\mathbb D$ whose point evaluation functionals are continuous. We study weighted composition operators from $X$ into Bloch type spaces. Imposing certain natural conditions on $X$ we are able to characterize all at once the bounded and the compact operators as well as in many cases give estimates or precise formulas for the essential norm. One condition used is: \medskip \noindent (VI) \ There exists $C\u3e0$ such that $\|Sf\|\le C\|f\|$, for all $f$ in $X$ and for all disk automorphisms $S$. \smallskip \noindent When $X$ is either the Bloch space or the space of analytic functions, $S^p$, whose derivatives are in the Hardy space $H^p$ though, (VI) fails. So when $X$ is continuously contained in the Bloch space, we impose two other conditions on the norm of the point evaluation functionals. In the end our results apply to known spaces that include the Hardy spaces, the weighted Bergman spaces, $BMOA$, the Besov spaces and all spaces $S^p$. This is joint work with Flavia Colonna

Closed Range Composition Operators on Hilbert Function Spaces

We show that a Carleson measure satisfies the reverse Carleson condition if and only if its Berezin symbol is bounded below on the unit disk D. We provide new necessary and sufficient conditions for the composition operator to have closed range on the Bergman space. The pull-back measure of area measure on D plays an important role. We also give a new proof in the case of the Hardy space and conjecture a condition in the case of the Dirichlet space

Closed Range Composition Operators on Hilbert Function Spaces

We show that a Carleson measure satisfies the reverse Carleson condition if and only if its Berezin symbol is bounded below on the unit disk D. We provide new necessary and sufficient conditions for the composition operator to have closed range on the Bergman space. The pull-back measure of area measure on D plays an important role. We also give a new proof in the case of the Hardy space and conjecture a condition in the case of the Dirichlet space

Closed-Range Composition Operators on A2 and the Bloch Space

For any analytic self-map φ of {z : |z| \u3c 1} we give four separate conditions, each of which is necessary and sufficient for the composition operator Cφ to be closed-range on the Bloch space B . Among these conditions are some that appear in the literature, where we provide new proofs. We further show that if Cφ is closed-range on the Bergman space A2 , then it is closed-range on B , but that the converse of this fails with a vengeance. Our analysis involves an extension of the Julia-Carathéodory Theorem

Closed-Range Composition Operators on A2 and the Bloch Space

For any analytic self-map φ of {z : |z| \u3c 1} we give four separate conditions, each of which is necessary and sufficient for the composition operator Cφ to be closed-range on the Bloch space B . Among these conditions are some that appear in the literature, where we provide new proofs. We further show that if Cφ is closed-range on the Bergman space A2 , then it is closed-range on B , but that the converse of this fails with a vengeance. Our analysis involves an extension of the Julia-Carathéodory Theorem