Let T = U｜T｜ be the polar decomposition of a bounded linear operator T on a Hilbert space. The transformation T = ｜T｜^1/2 U｜T｜^1/2 is called the Aluthge transformation and Tn means the n-th Aluthge transformatio...Let T = U｜T｜ be the polar decomposition of a bounded linear operator T on a Hilbert space. The transformation T = ｜T｜^1/2 U｜T｜^1/2 is called the Aluthge transformation and Tn means the n-th Aluthge transformation. Similarly, the transformation T（＊）=｜T＊｜^1/2 U｜T＊｜＆1/2 is called the ＊-Aluthge transformation and Tn^（＊） means the n-th ＊-Aluthge transformation. In this paper, firstly, we show that T（＊） = UV｜T^（＊）｜ is the polar decomposition of T（＊）, where ｜T｜^1/2 ｜T^＊｜^1/2 = V｜｜T｜^1/2 ｜T^＊｜^1/2｜ is the polar decomposition. Secondly, we show that T（＊） = U｜T^（＊）｜ if and only if T is binormal, i.e., [｜T｜, ｜T^＊｜]=0, where [A, B] = AB - BA for any operator A and B. Lastly, we show that Tn^（＊） is binormal for all non-negative integer n if and only if T is centered, and so on.展开更多
In this note an alternative proof of the equivalence of Drazin invertibility of operators AB and BA is given. As an application, we will prove that σD(AB) = σD(BA) and σD(A) = σD(A), where σD(M) and ■ denote the...In this note an alternative proof of the equivalence of Drazin invertibility of operators AB and BA is given. As an application, we will prove that σD(AB) = σD(BA) and σD(A) = σD(A), where σD(M) and ■ denote the Drazin spectrum and the Aluthge transform of an operator M ∈ B(H), respectively.展开更多
Let X, Y be Banach spaces, R ： X → Y emd S ： Y →X be bounded linear operators. When λ ≠ 0, we investigate common properties of λ i - SR and ,λ I - RS. This work should be viewed as a continuation of researches...Let X, Y be Banach spaces, R ： X → Y emd S ： Y →X be bounded linear operators. When λ ≠ 0, we investigate common properties of λ i - SR and ,λ I - RS. This work should be viewed as a continuation of researches of Barnes and Lin et al.. We also apply these results obtained to B-Fredholm theory, extensions and Aluthge transforms.展开更多
基金Science Foundation of Minisitry of Education of China (No.208081)
文摘Let T = U｜T｜ be the polar decomposition of a bounded linear operator T on a Hilbert space. The transformation T = ｜T｜^1/2 U｜T｜^1/2 is called the Aluthge transformation and Tn means the n-th Aluthge transformation. Similarly, the transformation T（＊）=｜T＊｜^1/2 U｜T＊｜＆1/2 is called the ＊-Aluthge transformation and Tn^（＊） means the n-th ＊-Aluthge transformation. In this paper, firstly, we show that T（＊） = UV｜T^（＊）｜ is the polar decomposition of T（＊）, where ｜T｜^1/2 ｜T^＊｜^1/2 = V｜｜T｜^1/2 ｜T^＊｜^1/2｜ is the polar decomposition. Secondly, we show that T（＊） = U｜T^（＊）｜ if and only if T is binormal, i.e., [｜T｜, ｜T^＊｜]=0, where [A, B] = AB - BA for any operator A and B. Lastly, we show that Tn^（＊） is binormal for all non-negative integer n if and only if T is centered, and so on.
基金the National Natural Science Foundation of China (No.10571113)
文摘In this note an alternative proof of the equivalence of Drazin invertibility of operators AB and BA is given. As an application, we will prove that σD(AB) = σD(BA) and σD(A) = σD(A), where σD(M) and ■ denote the Drazin spectrum and the Aluthge transform of an operator M ∈ B(H), respectively.
基金Supported by National Natural Science Foundation of China(Grant No.11171066)Special Funds of National Natural Science Foundation of China(Grant No.11226113)+2 种基金Specialized Research Fund for the Doctoral Program of Higher Education(Grant Nos.2010350311001 and 20113503120003)Natural Science Foundation of Fujian Province(Grant Nos.2011J05002 and 2012J05003)Foundation of the Education Department of Fujian Province(Grant No.JB10042)
文摘Let X, Y be Banach spaces, R ： X → Y emd S ： Y →X be bounded linear operators. When λ ≠ 0, we investigate common properties of λ i - SR and ,λ I - RS. This work should be viewed as a continuation of researches of Barnes and Lin et al.. We also apply these results obtained to B-Fredholm theory, extensions and Aluthge transforms.
