Let H be a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator A in L(H) is said to be a Cowen-Douglas operator if there exist , a connected open subset of complex plane C, and n, a positive integer, such thatIn the paper, we give a similarity classification of Cowen-Douglas operators by using the ordered K-group of the commutant algebra as an invariant, and characterize the maximal ideals of the commutant algebras of Cowen-Douglas operators. The theorem greatly generalizes the main result in (Canada J. Math. 156(4) (2004) 742) by simply removing the restriction of strong irreducibility of the operators. The research is also partially inspired by the recent classification theory of simple AH algebras of Elliott-Gong in (Documenta Math. 7 (2002) 255; ଁ

No abstract

The (U + K)-orbit of a bounded linear operator T acting on a Hilbert space H is defined as (U + K)(T )={R −1 T R: R is invertible of the form unitary plus compact on H}. In this paper, we first characterize the closure of the (U + K)-orbit of an essentially normal triangular operator T satisfying H = {ker(T −λI) : λ ∈ ρ F (T )} and σp(T * ) = ∅. After that, we establish certain essentially normal triangular operator models with the form of the direct sums of triangular operators, adjoint of triangular operators and normal operators, show that such operator models generate the same closed (U + K)-orbit if they have the same spectral picture, and describe the closures of the (U + K)-orbits of these operator models. These generalize some known results on the closures of (U + K)-orbits of essentially normal operators, and provide more positive cases to an open conjecture raised by Marcoux as Question 2 in his article "A survey of (U + K)-orbits".

No abstract

Let 𝓗 be a complex separable Hilbert space and ℒ(𝓗) denote the collection of bounded linear operators on 𝓗. In this paper, we show that for any operator A ∈ ℒ(𝓗), there exists a stably finitely (SI) decomposable operator A∈, such that ‖A−A∈‖ 𝓗 ∈ andA′(A∈)/ rad A′(A∈) is commutative, where rad A′(A∈) is the Jacobson radical of A′(A∈). Moreover, we give a similarity classification of the stably finitely decomposable operators that generalizes the result on similarity classification of Cowen–Douglas operators given by C. L. Jiang.

In many practical problems, it is sometimes necessary to evaluate the derivative of function whose values are given approximately. Firstly, the problem of estimating the derivative of a function observed with error is studied. It presents a proper regularization strategy and explain how to choose regularization parameter. Secondly, the regularization strategy above to the numerical differentiation is applied and discussed in the implementation of the numerical method and the tests which it has performed in order to investigate the accuracy and stability of the numerical differentiation procedure. Finally, some numerical examples will further illustrate that this method is reasonable, effective and reasonable.