Questions and Answers (3) View all
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Answer added in Real and Complex Analysis13 2-quasinormal operators, any ideas or applications?Dear Dan Sorry, It was my mistake. I did not see T in first line. Anyway, suppose T is invertible, from T(T*T)=2((T*T)T), we can deduce that TT*=2T*T.... [more]
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Answer added in Real and Complex Analysis13 2-quasinormal operators, any ideas or applications?Dear Dan (T*T)=2((T*T)T)====> T*T-2(T*T)T=0====> T*T(1-2T)=0. Here 1 is the identity operator.
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Answer added in Real and Complex Analysis13 2-quasinormal operators, any ideas or applications?Dear Dan If T Is 2-quasinormal, then T*T(I-2T)=0. Threfore, T*T is not invertible. Hence T is not invertible.
Publications (2) View all
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Article: Homomorphisms of certain Banach function algebras
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ABSTRACT: In this note, we study homomorphisms with domainD n(X) orLipα(X, d) of which ranges are certain Banach function algebras and determine in which cases these homomorphisms are compact.Proceedings Mathematical Sciences 04/2012; 112(2):331-336. · 0.17 Impact Factor -
Article: Compact endomorphisms of certain analytic Lipschitz algebras
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ABSTRACT: Let $X$ be a compact plane set. $A(X)$ denotes the uniform algebra of all continuous complex-valued functions on $X$ which are analytic on int$X$. For $0<\alpha\leq 1$, Lipschitz algebra of order $\alpha$, $Lip(X,\alpha)$ is the algebra of all complex-valued functions $f$ on $X$ for which $p_\alpha(f)=\sup\{\frac{|f(x)-f(y)|}{|x-y|^{\alpha}}:x,y\in X, x\neq y\}<\infty.$ Let $Lip_A(X,\alpha)=A(X)\bigcap Lip(X,\alpha)$, and $Lip^{n}(X,\alpha)$ be the algebra of complex-valued functions on $X$ whose derivatives up to order $n$ are in $\Lip(X,\alpha)$. $Lip_A(X,\alpha)$ under the norm $\|f\|=\|f\|_X+p_\alpha(f)$, and $Lip^n(X,\alpha)$ for a certain plane set $X$ under the norm $\|f\|=\sum_{k=0}^{n}\frac{\|f^{(k)}\|_X+p_{\alpha}(f^{(k)})}{k!}$ are natural Banach function algebras, where $\|f\|_X = \sup_{x\in X } |f(x)|$. In this note we study endomorphisms of algebras $Lip_A(X,\alpha)$ and $Lip^n(X,\alpha)$ and investigate necessary and sufficient conditions for which these endomorphisms to be compact. Finally, we determine the spectra of compact endomorphisms of these algebras.
