## On Unbounded Bergman Operators |

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### Contents

Three Extension of the BergerShaw Theorem | 15 |

Four Commutants of Bergman operator | 34 |

Five Open problems | 43 |

### Common terms and phrases

Area G B"f e Berger-Shaw Theroem Bergman space bounded analytic function bounded Bergman operator bounded extension bounded linear functional bounded linear operator bounded open subset bounded operator bounded real Cartesian decomposition Cauchy sequence closed densely defined closed symmetric operator conformal map decomposition with bounded dense in L’,(G dense subset densely defined operator dim ker Doctor of Philosophy domain extend the Berger-Shaw extension of spectral f e H F. H. Szafraniec finite G z+o Hilbert space Hilbert-Schmidt operator iB)h isometric isomorphism L2 G Let f e linear span minimal normal extension multiplication open problem operator for G operators commuting orthonormal basis pf for f positive symmetric operator Proof self-adjoint operator space of square spectral type spectrum square integrable functions Stochel and F. H. subset G sufficient condition summand Theorem 4.4 Theorem Let trace class operator unbounded subnormal operators whole plane z e G