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Composition and multiplication operators on the derivative Hardy space \(S^2(\mathbb D)\). (English) Zbl 1500.47037

Summary: In this paper, we propose a different (and equivalent) norm on \(S^2(\mathbb D)\) which consists of functions whose derivatives are in the Hardy space of unit disk. The reproducing kernel of \(S^2(\mathbb D)\) in this norm admits an explicit form, and it is a complete Nevanlinna-Pick kernel. Furthermore, there is a surprising connection of this norm with 3-isometries. We then study composition and multiplication operators on this space. Specifically, we obtain an upper bound for the norm of \(C_\varphi\) for a class of composition operators. We completely characterize multiplication operators which are \(m\)-isometries. As an application of the 3-isometry, we describe the reducing subspaces of \(M_\varphi\) on \(S^2(\mathbb D)\) when \(\varphi\) is a finite Blaschke product of order 2.

MSC:

47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
47B33 Linear composition operators
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
30H10 Hardy spaces
30H20 Bergman spaces and Fock spaces
30H50 Algebras of analytic functions of one complex variable
30J10 Blaschke products
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References:

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