Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Multiplication operators on the Bergman space via analytic continuation   In this paper, using the group-like property of local inverses of a finite
Blaschke product $\phi$, we will show that the largest $C^*$-algebra in the
commutant of the multiplication operator $M_{\phi}$ by $\phi$ on the Bergman
space is finite dimensional, and its dimension equals the number of connected
components of the Riemann surface of $\phi^{-1}\circ\phi $ over the unit disk.
If the order of the Blaschke product $\phi$ is less than or equal to eight,
then every $C^*$-algebra contained in the commutant of $M_{\phi}$ is abelian
and hence the number of minimal reducing subspaces of $M_{\phi}$ equals the
number of connected components of the Riemann surface of $\phi^{-1}\circ\phi $
over the unit disk.

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The categories of this paper are: econ
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The categories of this paper are: funct-an
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: cs
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The categories of this paper are: supr-con
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The categories of this paper are: alg-geom
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The categories of this paper are: stat
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The categories of this paper are: gr-qc
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: q-fin
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The categories of this paper are: eess
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The categories of this paper are: physics