On the injective norm and characterization of some subclasses of normal operators by inequalities or equalities

Let B(H) be the C^*-algebra of all bounded linear operators acting on a complex Hilbert space H. In this note, we shall show that if S is an invertible normal operator in B(H) the following estimation holds

     

Near Subnormality of Weighted Shifts and the Answer to the Hilbert Space Problem 160

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　Ｉｎ [1 ] ,ｔｗｏｏｐｅｒａｔｏｒｃｌａｓｓｅｓｏｆＤ ｎｅａｒｓｕｂｎｏｒｍａｌｏｐｅｒａｔｏｒｓａｎｄｎｅａｒｓｕｂｎｏｒｍａｌｏｐｅｒａｔｏｒｓｗｅｒｅｉｎｔｒｏｄｕｃｅｄ .ＴｈｅｅｌｅｍｅｎｔａｒｙｐｒｏｐｅｒｔｉｅｓｏｆｓｕｃｈｏｐｅｒａｔｏｒｓｗｅｒｅｏｂｔａｉｎｅｄｂｙｕｓｅｏｆｔｈｅＭ Ｐｇｅｎｅｒａｌｉｚｅｄｉｎｖｅｒｓｅ.Ｍｏｒｅｏｖｅｒ,ａｎｅｗｎｅｃｅｓｓａｒｙａｎｄｓｕｆｆｉｃｉｅｎｔｃｏｎｄｉｔｉｏｎｆｏｒａｎｏｐｅｒａｔｏｒｔｏｂｅｓｕｂｎ…     

广义弱亚正规算子的正规性

主要研究了T是广义弱亚正规算子时,T~t是拟正规算子的充要条件是T是拟正规算子.并且举例说明了存在非次正规的广义弱亚正规算子T使得T~t是次正规的.     

良序套张量积的代数中的强不可约算子

设Ｈ为复的可分无限维Ｈｉｌｂｅｒｔ空间,称有界线性算子Ｔ为强不可约的,如果与Ｔ可交换的幂等算子只有０和Ｉ．王宗尧、蒋春澜、纪有清等人证明了在任何一个套的套代数中都存在大量的强不可约算子,并且找到了它们的酉轨道闭包．本文考虑有限个套的张量积的代数中强不可约算子的存在性问题。证明了：对复平面上任何一个连通完备集σ、总存在一个对角算子Ｎ和它的一个范数可以任意小的紧摄动Ｔ＝Ｘ＋Ｋ,使得Ｔ是一个强不可约算子、Ｔ在有限个良序套的张量积的代数中,并且σ（Ｔ）＝σｌｒｅ（Ｔ）＝σ（Ｎ）＝σｌｒｅ（Ｎ）＝σ进一步,文章还对具有单点谱的算子和良序套与正交补为良序套的张量积的代数进行了讨论,得到了一些结果．     

Special uniform approximations of continuous vector-valued functions. Part II: special approximations in CX(T)CY(S)

In this paper, which is a continuation of Timofte (J. Approx. Theory 119 (2002) 291–299, we give special uniform approximations of functions from C_XY(T×S) and C_∞(T×S,XY) by elements of the tensor products C_X(T)C_Y(S), respectively C₀(T,X)C₀(S,Y), for topological spaces T,S and Γ-locally convex spaces X,Y (all four being Hausdorff).     

加权移位算子是次正常算子的条件

以广义逆为工具运用算子演算给出加权移位算子是次正常算子的条件,所用方法不同于 Ｓｔａｍｐｆｌｉ的工作,但结果一致．作为应用给出了两个例子．     

Additive multiplicative increasing functions on nonnegative square matrices and multidigraphs

It is known that if f is a multiplicative increasing function on , then either f(n)=0 for all or f(n)=n for some 0. It is very natural to ask if there are similar results in other algebraic systems. In this paper, we first study the multiplicative increasing functions over nonnegative square matrices with respect to tensor product and then restrict our result to multidigraphs and loopless multidigraphs.     

关于一般矩阵函数变差的不等式

该文首先展示了一般矩阵函数dχ(A)=∑[DD(X]σ∈H[DD)]χ(σ)∏[DD(]m[]i=1[DD)]aiσ(i)可作为一个酉空间张量的适当对称类的内积.然后,借助Schwarz不等式和范数证明了关于一般矩阵函数变差的三个主要不等式,而其中一个不等式是已知不等式的推广.     

On quasi-similarity of subnormal operators

For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R2(K,v) denote the closure of Rat(K) in L2(v) and let Sv denote the operator of multiplication by the independent variable z on R2(K, v), that is, Svf = zf for every f∈R2(K, v). SupposeΩis a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H2(Ω). Letσdenote a harmonic measure forΩ. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R2(Ω,σ). We show that for a given v supported onΩ, Sv is quasi-similar to Sσif and only if v/■Ω■σ and log(dv/dσ)∈L1(σ). Our result extends a well-known result of Clary on the unit disk.