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Multipliers and invariant subspaces in the Dirichlet space. (English) Zbl 0810.46057

Let \(D\) denote the Dirichlet space of all analytic functions \(f\) on the open unit disc \(\Delta\) that have a finite Dirichlet integral, that is, \[ D(f)= \iint_ \Delta | f'(z)| dA(z)< \infty, \] where \(dA(z)\) denotes the normalized area measure on \(\Delta\). Let \((M_ z,D)\) denote the Dirichlet shift, i.e., \(M_ z\) is the operator of multiplication by \(z\) on the Dirichlet space \(D\). In this interesting paper the authors study the invariant subspace structure of \((M_ z,D)\). One of the main results proved is that every invariant subspace \(\mathcal M\) is generated by a multiplier \(\varphi\), that is, there is an analytic function \(\varphi\) on \(\Delta\) such that \(\varphi f\in D\) whenever \(f\in D\).

MSC:

46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
47A15 Invariant subspaces of linear operators
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