The Research On Several Properties For Numerical Range And Numerical Radius Of Bounded Linear Operators

Spectral theory,numerical range and generalized numerical range of linear operators are the active research topics in operator theory,which have profound theoretical foundation and wide application value.The research of spectral theory,numerical range and generalized numerical range has related to functional analysis,Banach algebras,inequalities,matrix polynomials and quantum physics,etc.Through the research for them,the interior relation of operator structure can be more clearly.Considering the important applications of spectral theory,numerical range and generalized numerical range in the above aspects,the classical numerical range in Hilbert space,the A-spectrum and A-numerical range in the semi-inner product space induced by positive operator A are mainly studied.The specific contents are as follows:Firstly,since the numerical radius is an effective tool to describe the range of numerical range,the classical numerical radius inequalities of 2 × 2 block operator matrices are studied.Several important inequalities in Hilbert space are introduced:the Grüss inequality,Jensen inequality,the Buzano type generalization of Schwarz inequality,etc.Combining these inequalities,the upper bounds of the classical numerical radius of the block operator matrices are described by the classical numerical radius of its entries.Particularly,some classical numerical radius inequalities of us generalize and refine the existing conclusions.Secondly,the characterizations of generalized spectrum(A-spectrum)and generalized numerical range(A-numerical range)in semi-inner product space induced by positive operator A are studied.Considering the interference of positive operator A in the inner product of this semi-inner product space,the methods for classical spectrum and classical numerical range will not be applied to A-spectrum and A-numerical range completely.For this reason,the classical spectrum and the classical numerical range in Hilbert space are used to describe the A-spectrum and A-numerical range.The conclusion provides an effective tool for further study of the A-spectrum and A-numerical range.Moreover,some basic properties of the A-spectrum and A-numerical range are obtained by these relationships.In particular,the A-spectrum inclusion property of A-numerical range can be verified from this equivalence characterization.Finally,the A-numerical radius inequalities in semi-inner product space induced by positive operator A are also studied.Some classical inequalities in Hilbert space are extended to the semi-inner product space.And the upper and lower bounds of Anumerical radius are obtained by using these inequalities,which are more accurate than the existing ones.Furthermore,a norm inequality involving powers of positive operators in Hilbert space is generalized to semi-inner product space,which obtains the A-norm inequality for A-positive operators.Some inequalities for A-numerical radius are also obtained via this A-norm inequality,which generalize the classical A-numerical radius inequalities.