Block Numerical Ranges And Estimable Decompositions Of Block Operator Matrices

This dissertation deals with a conjecture posed by Abbas Salemi in 2011（Banach J.Math.Anal.）,claiming that for every bounded linear operator,A on a separable Hilbert space there exists an estimable decomposition,such that σ（A）=∩_n=1^∞（?）.We investigate block numerical ranges of bounded linear operators on a separable Hilbert space and partially solve this problem.However,in the process of studying Salemi’s Conjecture,the existence on the estimable decomposition is,in general,hard to obtain and numerical approximations for the spectra may not be reliable,in particular,if the operator is not self-adjoint or normal.This dissertation arose from an attempt to gain a better understanding of the block numerical range and study Salemi’s Conjecture.And then we can get the related information of the spectrum of the operator.We consider how to compute block numerical range using projection methods,which can reduce the problem to that of computing the block numerical range of a（finite）block matrix.Firstly,we assert that Salemi’s Conj ecture holds if the operator is the following cases:diagonal operator;normal operator;hyponormal operator with the totally disconnected spectrum,some classes of hyponormal operators;a seminormal operator with the totally disconnected spectrum,and a certain class of seminormal operator.We prove that there exists an estimable decomposition for the aforementioned operators.Secondly,we assert that Salemi’s Conjecture holds if the operator is a nilpotent operator or a certain class of spectral operators.In the sense of norm-limit,the estimable decomposition of the quasi-nilpotent operator is obtained.Furthermore,this conjecture also holds if the operator is a spectral operator under quasi-nilpotent equivalence.Finally,we obtain an approximation of the block numerical range of bounded and unbounded block operator matrices using projection methods.When the block opera-tor matrix is unbounded,we do assume either that operator is diagonally dominant or off-diagonally dominant.As a simple example,we calculate approximately the quartic numerical range of a concrete infinite dimensional Hamiltonian operator matrix.