Numerical Ranges Of Special Operators On Hilbert Space

The theories of quadratic forms and their applications appear in many parts of mathematics and the sciences.This theories and its extensions to infinite dimensions comprise the theoty of the numerical range.Let H be a complex Hilbert space,with inner product denoted by(.,.),denote by L(H)the set of all bounded linear operators on H,and let T ∈ L(H).The numerical range W{T)of T is defined by the following equation:W(T)={(Tx,x)|x∈H,||x||=1}When H=Cn and A∈Mn(C)is an W(A)={x*Ax|x∈n,||x||=1},The following properties of W(A)are immediate:W{aI+bA)= a + bW+(A),for a,b∈C.W(A*)={λ|λ∈W(A)}W(U*AU)=W(A),for unitary U.It is well known that the numerical range of operator acting on a two-dimensional space is an elliptical disc with foci the eigenvalue of the operator(the Ellipse Lemma).By the Ellipse Lemma,the numerical range of any operator is immediately seen to be convex(the Toeplitz-Hausdorff Theorem).If the operator is acting on a finite-dimensional Hilbert space,then the numerical is in addition compact.The numerical radius w(A)of an operator A is defined as follows:w(A)= sup{|λ||λ∈W(A)}.In this paper,the first chapter reviews the research history and status quo of numerical ranges;The second chapter introduces the theory of operator,the definition and related properties of special operators and number range;The third chapter first section mainly introduced the 2 × 2 matrix numerical range and classical theorem-elliptical lemma,and its calculation and the geometric characteristics of numerical range have a preliminary understanding.The second section of this chapter mainly introduces euivalent conditions for W(A)to be a circular disk centered at the origin and upper triangular matrix numberical range of relevant research results.Third section of this chapter discusses the 4-square nilpotent matrix numerical range problems.According the studies for the finite dimensional nilpotent matrix numerical range,an equivalent condition can be given;In chapter 4,classic operators-self adjoint operator and normal operator that its related conclusions are summarized,and the numberical range and numerical radius of shift operators are studied,thus,the numerical radius of nilpotent operator are obtained.In addition,this paper introduces the upper and lower bounds of the block shift operator with numerical range.For the equivalent conditions of 3-square nilpotent matrix in real number field that its numberical range to be a disk,Marvin Marcus and Claire Pesce have given as early as 1986.This paper introduces the more general conclusion in complex number filed of third-order and fourth-order nilpotent matrix of numerical range to be circular disk and the calculation of the numerical radius:Theorem3.2.2 Let A=(aij)be an 3-square complex nilpotent upper triangular matrix,then the following conditions are equivalent:(1)W(A)is a circular disk centered at the origin.(2)a12a23a13 = 0.(3)tr(A2A*)=0,and then(?).Theorem3.3.1 Let A =(aij)be an 4-square complex nilpotent upper triangular matrix,then the following conditions are equivalent:(1)W(A)is a circular disk centered at the origin.(2)(?)and a12a23a34a14=0.(3)tr(A2A*)=0 and tr(A3A*)=0,then w(A)=(?)The fourth chapter summarizes the numerical radius of nilpotent operator and block shift operator inequality:Theorem4.3.2 Let T∈L(H),Tn=0,n≥2,then(1)(?).(2)Suppose moreover that(?),Suppose that there exists a unit vector(?)Let Vξ= Span{ξ,Tξ,T2ξ,…,Tn-1ξ},Then Vξ is an n-dismensional subspace of H which is reducing for T,and the restriction of T to Vξ is unitarily equivalent to the n-dismensional shift on Cn.Theorem4.4.1 Let(?)on(?),be an n-by-n block shift,where Tj is nj × nj+1(1≤j≤k-1)matrix,and let(?)on Ck,Express T =(?),where Aj,Bj is either a zero matrix or of the form(?)Then(1)w(T)≤w((?))(2)w(T)=w((?))if and only if T is unitarily similar to(?),1≤j0≤m，where j0 is such that(?)，where C is a shift block with w(C)≤w(Bj0).(3)Let Tj≠0,we have w(T)= w((?))if and only if T is unitary similar to(?),where C is a block shift with w(C)≤w((?)).