## 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 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 dense in H dense subset densely defined operator dim ker(B Doctor of Philosophy domain extension of spectral f e L G f e L2 F. H. Szafraniec finite function on G Hilbert space Hilbert-Schmidt operator isometric isomorphism ker(A L2 G Lemma Let f e linear span minimal normal extension multiplication n n n=l n=l n=l open problem operator for G operator on H operators commuting orthonormal basis positive symmetric operator Proof self-adjoint operator self-commutator S*,S space of square spectral type spectrum square integrable functions subset G sufficient condition Theorem 4.4 Theorem Let trace class operator unbounded subnormal operators whole plane z e G zf for f