| Abstract | 第1-6页 |
| 摘要 | 第6-10页 |
| 1 Introduction | 第10-22页 |
| ·Background | 第10-17页 |
| ·Block operator matrices | 第10-13页 |
| ·Quadratic numerical range and spectral properties | 第13-15页 |
| ·Completion problems | 第15-17页 |
| ·utline of this work | 第17-19页 |
| ·Notations | 第19-22页 |
| 2 Quadratic numerical range | 第22-36页 |
| ·Preliminaries | 第22-23页 |
| ·Fundamental properties of quadratic numerical range | 第23-27页 |
| ·Block numerical range | 第27-36页 |
| 3 Spectral inclusion properties | 第36-58页 |
| ·Spectral inclusion properties of unbounded Hamiltonian operators | 第36-47页 |
| ·Preliminaries | 第37-38页 |
| ·Off-diagonally Hamiltonian case | 第38-39页 |
| ·Upper triangular Hamiltonian case | 第39-42页 |
| ·General 2×2 Hamiltonian case | 第42-47页 |
| ·Spectral inclusion properties of some unbounded block operator matrices | 第47-58页 |
| ·Preliminaries | 第47-49页 |
| ·Hypo-normal diagonal entry case | 第49-50页 |
| ·General diagonal entry case | 第50-52页 |
| ·The case of C =±B~* | 第52-58页 |
| 4 Invertibility 0f symplectic symmetric block operator matrices | 第58-72页 |
| ·Preliminaries | 第59-60页 |
| ·Invertibility of symplectic symmetric operator matrices | 第60-72页 |
| ·Symplectic symmetric operator matrix with inf (?) r>0 | 第61-63页 |
| ·Non-negative symplectic symmetric operator matrix | 第63-65页 |
| ·Symplectic symmetric operator matrices with diagonal (off-diagonal)domain | 第65-68页 |
| ·Invertibility of symplectic self_adjoint operator matrices | 第68-72页 |
| 5 Completion problems of unbounded triangular operator matrices | 第72-90页 |
| ·Closedness of ranges of unbounded upper triangular operator ma-trices | 第72-83页 |
| ·Auxiliary propositions | 第73-79页 |
| ·Main results and proofs | 第79-83页 |
| ·Left invertible completions of unbounded upper triangular opera-tor matrices | 第83-90页 |
| ·Main results and proofs | 第83-89页 |
| ·An Example | 第89-90页 |
| 6 Completion problems of formal Hamiltonian operators | 第90-108页 |
| ·Left invertible completion of formal Hamiltonian operators | 第91-98页 |
| ·Upper triangular case | 第91-95页 |
| ·General 2×2 case | 第95-98页 |
| ·Invertibility and invertible comletion of formal Hamiltonian op-erators | 第98-108页 |
| ·Upper triangular case | 第99-103页 |
| ·General 2×2 case | 第103-106页 |
| ·An example | 第106-108页 |
| Conclusions | 第108-110页 |
| Bibliography | 第110-120页 |
| List of Symbols | 第120-122页 |
| Acknowledgements | 第122-124页 |
| List of publications and scholarship | 第124-125页 |
