Description of invariant subspaces in terms of Berezin symbols


Abstract: We consider the stretching operator $\left( T_{w}f\right) \left( z\right) =f(wz)$ and the multiple shift operator $S^{n}f=z^{n}f$ on the Hardy spaces $% H^{p}(\mathbb{D})$ $\left( 1\leq p<+\infty \right) .$ We describe in terms of so-called Berezin symbols their lattice of invariant subspaces. We also define a new class of operators on the reproducing kernel Hilbert space $% H(\Omega ),$ which in a particular case contains all compact operators, and discuss in terms of Berezin symbols their invariant subspaces.

Keywords: Invariant subspaces, multiple shift operator, Berezin symbol, stretching operator

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