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A Primer on the Dirichlet Space by Thomas Ransford, Javad Mashreghi, Karim Kellay, Omar El-Fallah

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5

Multipliers

The Dirichlet space image is not an algebra. We saw this in §1.3, where we also proved that image is an algebra, even a Banach algebra with respect to a suitable norm. This is reminiscent of the case of Hardy spaces: H2 is not an algebra, but H is one. However, there is also a significant difference between the Dirichlet and Hardy cases. The algebra H derives much of its importance from the fact that its elements are multipliers of H2, in other words, that

On the other hand, it turns out that not all elements of ∩ H are multipliers of

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