The Research On Generalized Numerical Range Of Block Operator Matrices

The study of numerical range started in 1918-1919 by Toeplitz and Hausdorff when they proved that the numerical range is always convex,since then,the study of numerical range,numerical radius and various generalized numerical range and their numerical ra-dius has attracted many scholars'attention.The subject is related and has applications to many different branches of pure and applied science such as operator theory,function-al analysis,~*algebras,Banach algebras,matrix norms,inequalities,numerical analysis,perturbation theory,systems theory and quantum physics,etc.In addition,their research methods are also very rich,algebra,analysis,geometry,com-binatorial theory,computer programming and so on are very useful research tools.Considering the important ap-plications of numerical range and generalized numerical range in the above aspects,we mainly study two types of generalized numerical range:quadratic numerical range and higher rank numerical range.The details are as follows.First,we introduced several important inequalities:generalized Schwartz inequality,Jensen inequality,Young inequality,Gr¨uss inequality.The upper bound of the norm of block operator matrix is given,and the obtained norm inequality is generalized,which lays a theoretical foundation on the connectivity of the quadratic numerical range and the estimation of the quadratic numerical radius.In addition,the norm of block operator matrix can also use to estimate the range of the numerical radius.Secondly,for bounded linear operators,the quadratic numerical range is a subset of the numerical range and the quadratic numerical range also has the property of spectral inclusion.Therefore,in terms of spectral characterization of linear operators,it can provide more accurate estimation information than the numerical range.However,the quadratic numerical range is not as convex as the numerical range,or even disconnected.Therefore,in this paper,several different conditions for judging the quadratic numerical range as a connected set and an unconnected set are given.As an application,several conditions are given that the quadratic numerical range of bounded Hamilton operator does not intersect the imaginary axis.In addition,for the numerical range,the operator is self-adjoint if and only if its numerical range is contained in the real axis,but for the quadratic numerical range,it is not so simple.Therefore,the sufficient and necessary condition for the quadratic numerical range to be contained in a straight line in the plane is given.As a corollary,the necessary and sufficient condition for the quadratic numerical range to be contained in the real axis or imaginary axis is given.Then,considering that the quadratic numerical radius of bounded linear operator is a powerful tool for characterizing the spectrum,we use the important inequalities given above to give the upper and lower bounds of the quadratic numerical radius of the off-diagonal block operator matrix.Finally,another generalized numerical range-higher rank numerical range is studied.As a tool of quantum error correction,higher rank numerical range has a profound appli-cation background.Therefore,in this paper,we mainly study the relationship between the convex hull of higher rank numerical range of the whole block operator matrix and that of its entries,and the inequality of higher rank numerical radius of special block operator matrix.In addition,we give the boundedness,convexity and connectivity of the higher rank numerical range of operator polynomial.