The Operator Matrix Of Volterra Intergral Operator

Let H be a Hilbert space and L（H）be the set of all bounded linear operators on H.For T ∈L（H）,the spectrum of T,denoted by σ（T）,is defined as the set of complex numbers λ for which λ-T is not invertible.The spectral structure of operators has always been an important topic in functional analysis and operator theory.If σ（T）={0},then T is called a quasinilpotent operator.Quasinilpotent operators have attracted much attention,and research on the power set,invariant subspaces,functional calculus,and other issues has yielded many excellent results.For T ∈ L（H）,if a closed subspace M of H satisfies TM （?） M,then M is called an invariant subspace of T.In this case,T is unitarily equivalent to the upper triangular matrix（A B C）,where A:M→M,B:M^⊥→M,and C:M^⊥→M^⊥.That is,the operator T can be represented by an upper triangular matrix.Let H=L²（0,1）and let V be the Volterra integral operator defined by Vf（x）=∫_x¹ f（t）dt for f ∈H.As a classical quasinilpotent operator,the Volterra operator V has long been of interest,and research on the norm of polynomials in V has been ongoing,with many significant results.However,the exact value of ‖Vⁿ‖ and the positive operator square root |Vⁿ| of Vⁿ remain unknown.We know that the invariant subspaces Nt=f∈H:f（x）=0,t ≤x≤1 of V form a decreasing chain.Therefore,V can be represented as an upper triangular matrix of any order.Further investigation of this matrix may provide additional insights for estimating ‖Vⁿ‖ and exploring |Vⁿ|.In this article,we represent the Volterra integral operator V as an N-th order operator matrix,calculate the specific expressions for the elements at any position in its n-th power,and estimate the norm of its n-th power elements when N=3.In Chapter 1,we introduced the research background of the Volterra integral operator and gave the concepts of the root numerical range and projection numerical range of the operator.In Chapter 2,we proved that V is unitarily equivalent to the operator matrix K on⊕_k=1^NH,where N is any positive integer,and where e₀（?）e₀ is a rank-one operator on H:e₀（?）e₀f=<f,e₀>e₀.Then,we calculated the specific expressions for the elements Kn（i,j）at the i-th row and j-th column of Kn.In Chapter 3,we estimated the operator norm of the elements of K when K is a third-order matrix,and studied some of its properties.