The trifecta of Hilbert spaces on the unit disc


Abstract: Hilbert spaces are common. Connection among them is rare. With this spirit, we will see the connection among the three classical Hilbert spaces on the unit disc U. These Hilbert spaces are Hardy ($H^2(U)$), Dirichlet ($D^2(U)$) and lastly Bergman ($A^2(U)$). The idea comes from the intuition of evolving the remarkable identity called as Littlewood Paley Identity for Bergman and Dirichlet spaces. During the presentation, audience will get to know more about this identity. And also, the importance of this identity in the sense that how it relates to the Nevanlinna Counting function. Nevanlinna Counting function $N_\phi(w)$ measures the ‘affinity’ that $\phi$ has for value $w$ where $\phi$ is a holomorphic map on $U$. Besides this, we will also see more connection among Hardy, Dirichlet and Riemann Zeta functions.

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