The Quadratic Numerical Range And Completion Problems Of Unbounded Operator Matrices

This work deals with some problems of unbounded block operator matrices in a Hilbert space including the quadratic numerical range, spectral inclusion properties, invertibility and completion problems.First, we study elementary properties of the quadratic numerical range of bounded block operator matrices. For example, for the block operator matrix A and for every unitary operator U, the union of quadratic numerical ranges of U*AU equals to the numerical range of A. Moreover, some results are generalized to unbounded cases immediately.Then, we use the quadratic numerical range and Gershgorin-type theorem to consider the spectral inclusion properties of some unbounded block operator ma-trices such as Hamiltonian and unbounded block operator matrices with bounded off-diagonal entries. In addition, the invertibilites of symplectic symmetric block operator matrices are further investigated.Finally, we research the closed range completion and left invertible completion of the unbounded upper triangular operator matrix Me=(?), and the left invertible completion, invertibility and invertible completion of the unbounded formal Hamiltonian operator matrix HB=(?) We propose a new method based on discussions for the dimension d(A) of R(A)⊥, which is different from the traditional viewpoint, and obtain some necessary and sufficient conditions for the above completion problems. In fact, some results are of interest even in bounded case.