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Annals of Functional Analysis
Annals of Functional Analysis
Medghalchi, Alireza; Ramezani, Ramin
Let $K$ be a hypergroup. The purpose of this article is to study the question of involutions on algebras $M(K)^{**}$ , $L(K)^{**}$ , and $L_{c}(K)^{**}$ . We show that the natural involution of $M(K)$ has the canonical extension to $M(K)^{**}$ if and only if the natural involution of $L(K)$ has the canonical extension to $L(K)^{**}$ . Also, we give necessary and sufficient conditions for $M(K)^{**}$ and $L(K)^{**}$ to admit an involution extending the natural involution of $M(K)$ when $K$ is left amenable. Finally, we find the necessary and sufficient conditions for $L_{c}(K)^{**}$ to admit an involution.
Medghalchi, Alireza; Ramezani, Ramin
Let $K$ be a hypergroup. The purpose of this article is to study the question of involutions on algebras $M(K)^{**}$ , $L(K)^{**}$ , and $L_{c}(K)^{**}$ . We show that the natural involution of $M(K)$ has the canonical extension to $M(K)^{**}$ if and only if the natural involution of $L(K)$ has the canonical extension to $L(K)^{**}$ . Also, we give necessary and sufficient conditions for $M(K)^{**}$ and $L(K)^{**}$ to admit an involution extending the natural involution of $M(K)$ when $K$ is left amenable. Finally, we find the necessary and sufficient conditions for $L_{c}(K)^{**}$ to admit an involution.
Medghalchi, Alireza; Ramezani, Ramin
Let $K$ be a hypergroup. The purpose of this article is to study the question of involutions on algebras $M(K)^{**}$ , $L(K)^{**}$ , and $L_{c}(K)^{**}$ . We show that the natural involution of $M(K)$ has the canonical extension to $M(K)^{**}$ if and only if the natural involution of $L(K)$ has the canonical extension to $L(K)^{**}$ . Also, we give necessary and sufficient conditions for $M(K)^{**}$ and $L(K)^{**}$ to admit an involution extending the natural involution of $M(K)$ when $K$ is left amenable. Finally, we find the necessary and sufficient conditions for $L_{c}(K)^{**}$ to admit an involution.
Hazarika, Munmun; Gogoi, Pearl S.
We introduce a family $\mathcal{T}$ consisting of invertible matrices with exactly one nonzero entry in each row and each column. The elements of $\mathcal{T}$ are possibly mutually noncommuting, and they need not be normal or self-adjoint. We consider an operator-valued unilateral weighted shift $W$ with a uniformly bounded sequence of weights belonging to $\mathcal{T}$ , and we describe its minimal reducing subspaces.
Hazarika, Munmun; Gogoi, Pearl S.
We introduce a family $\mathcal{T}$ consisting of invertible matrices with exactly one nonzero entry in each row and each column. The elements of $\mathcal{T}$ are possibly mutually noncommuting, and they need not be normal or self-adjoint. We consider an operator-valued unilateral weighted shift $W$ with a uniformly bounded sequence of weights belonging to $\mathcal{T}$ , and we describe its minimal reducing subspaces.
Hazarika, Munmun; Gogoi, Pearl S.
We introduce a family $\mathcal{T}$ consisting of invertible matrices with exactly one nonzero entry in each row and each column. The elements of $\mathcal{T}$ are possibly mutually noncommuting, and they need not be normal or self-adjoint. We consider an operator-valued unilateral weighted shift $W$ with a uniformly bounded sequence of weights belonging to $\mathcal{T}$ , and we describe its minimal reducing subspaces.
Cobos-Sánchez, Clemente; García-Pacheco, Francisco Javier; Moreno-Pulido, Soledad; Sáez-Martínez, Sol
The set of supporting vectors of a continuous linear operator, that is, the normalized vectors at which the operator attains its norm, is decomposed into its convex components. In the complex case, the set of supporting vectors of a nonzero functional is proved to be path-connected. We also introduce the concept of generalized supporting vectors for a sequence of operators as the normalized vectors that maximize the summation of the squared norm of those operators. We determine the set of generalized supporting vectors for the particular case of a finite sequence of real matrices. Finally, we unveil the relation between...
Cobos-Sánchez, Clemente; García-Pacheco, Francisco Javier; Moreno-Pulido, Soledad; Sáez-Martínez, Sol
The set of supporting vectors of a continuous linear operator, that is, the normalized vectors at which the operator attains its norm, is decomposed into its convex components. In the complex case, the set of supporting vectors of a nonzero functional is proved to be path-connected. We also introduce the concept of generalized supporting vectors for a sequence of operators as the normalized vectors that maximize the summation of the squared norm of those operators. We determine the set of generalized supporting vectors for the particular case of a finite sequence of real matrices. Finally, we unveil the relation between...
Cobos-Sánchez, Clemente; García-Pacheco, Francisco Javier; Moreno-Pulido, Soledad; Sáez-Martínez, Sol
The set of supporting vectors of a continuous linear operator, that is, the normalized vectors at which the operator attains its norm, is decomposed into its convex components. In the complex case, the set of supporting vectors of a nonzero functional is proved to be path-connected. We also introduce the concept of generalized supporting vectors for a sequence of operators as the normalized vectors that maximize the summation of the squared norm of those operators. We determine the set of generalized supporting vectors for the particular case of a finite sequence of real matrices. Finally, we unveil the relation between...
Chen, Dongyang
Let $X,Y$ be Banach spaces. We define \begin{equation*}\alpha_{Y}(X)=\sup\{\vert T^{-1}\vert^{-1}:T:Y\rightarrow X\mbox{ is an isomorphism with }\vert T\vert \leq1\}.\end{equation*} If there is no isomorphism from $Y$ to $X$ , we set $\alpha_{Y}(X)=0$ , and ¶ \begin{equation*}\gamma_{Y}(X)=\sup\{\delta(T):T:X\rightarrow Y\mbox{ is asurjective operator with }\vert T\vert \leq1\},\end{equation*} where $\delta(T)=\sup\{\delta\gt 0:\delta B_{Y}\subseteq TB_{X}\}$ . If there is no surjective operator from $X$ onto $Y$ , we set $\gamma_{Y}(X)=0$ . We prove that for a separable space $X$ , $\alpha_{l_{1}}(X^{*})=\gamma_{c_{0}}(X)$ and $\alpha_{L_{1}}(X^{*})=\gamma_{C(\Delta)}(X)=\gamma_{C[0,1]}(X)$ .
Chen, Dongyang
Let $X,Y$ be Banach spaces. We define \begin{equation*}\alpha_{Y}(X)=\sup\{\vert T^{-1}\vert^{-1}:T:Y\rightarrow X\mbox{ is an isomorphism with }\vert T\vert \leq1\}.\end{equation*} If there is no isomorphism from $Y$ to $X$ , we set $\alpha_{Y}(X)=0$ , and ¶ \begin{equation*}\gamma_{Y}(X)=\sup\{\delta(T):T:X\rightarrow Y\mbox{ is asurjective operator with }\vert T\vert \leq1\},\end{equation*} where $\delta(T)=\sup\{\delta\gt 0:\delta B_{Y}\subseteq TB_{X}\}$ . If there is no surjective operator from $X$ onto $Y$ , we set $\gamma_{Y}(X)=0$ . We prove that for a separable space $X$ , $\alpha_{l_{1}}(X^{*})=\gamma_{c_{0}}(X)$ and $\alpha_{L_{1}}(X^{*})=\gamma_{C(\Delta)}(X)=\gamma_{C[0,1]}(X)$ .
Chen, Dongyang
Let $X,Y$ be Banach spaces. We define \begin{equation*}\alpha_{Y}(X)=\sup\{\vert T^{-1}\vert^{-1}:T:Y\rightarrow X\mbox{ is an isomorphism with }\vert T\vert \leq1\}.\end{equation*} If there is no isomorphism from $Y$ to $X$ , we set $\alpha_{Y}(X)=0$ , and ¶ \begin{equation*}\gamma_{Y}(X)=\sup\{\delta(T):T:X\rightarrow Y\mbox{ is asurjective operator with }\vert T\vert \leq1\},\end{equation*} where $\delta(T)=\sup\{\delta\gt 0:\delta B_{Y}\subseteq TB_{X}\}$ . If there is no surjective operator from $X$ onto $Y$ , we set $\gamma_{Y}(X)=0$ . We prove that for a separable space $X$ , $\alpha_{l_{1}}(X^{*})=\gamma_{c_{0}}(X)$ and $\alpha_{L_{1}}(X^{*})=\gamma_{C(\Delta)}(X)=\gamma_{C[0,1]}(X)$ .
Meng, Qing
We define and study the weak Haagerup property for $C^{*}$ -algebras in this article. A $C^{*}$ -algebra with the Haagerup property always has the weak Haagerup property. We prove that a discrete group has the weak Haagerup property if and only if its reduced group $C^{*}$ -algebra also has that property. Moreover, we consider the permanence of the weak Haagerup property under a few canonical constructions of $C^{*}$ -algebras.
Meng, Qing
We define and study the weak Haagerup property for $C^{*}$ -algebras in this article. A $C^{*}$ -algebra with the Haagerup property always has the weak Haagerup property. We prove that a discrete group has the weak Haagerup property if and only if its reduced group $C^{*}$ -algebra also has that property. Moreover, we consider the permanence of the weak Haagerup property under a few canonical constructions of $C^{*}$ -algebras.
Meng, Qing
We define and study the weak Haagerup property for $C^{*}$ -algebras in this article. A $C^{*}$ -algebra with the Haagerup property always has the weak Haagerup property. We prove that a discrete group has the weak Haagerup property if and only if its reduced group $C^{*}$ -algebra also has that property. Moreover, we consider the permanence of the weak Haagerup property under a few canonical constructions of $C^{*}$ -algebras.
Mukhamedov, Farrukh; Fadillah Embong, Ahmad; Rosli, Azizi
In the present paper, we consider cubic stochastic operators, and prove that the surjectivity of such operators is equivalent to their orthogonal-preserving property. In the last section we provide a full description of orthogonal-preserving (respectively, surjective) cubic stochastic operators on the $2$ -dimensional simplex.
Mukhamedov, Farrukh; Fadillah Embong, Ahmad; Rosli, Azizi
In the present paper, we consider cubic stochastic operators, and prove that the surjectivity of such operators is equivalent to their orthogonal-preserving property. In the last section we provide a full description of orthogonal-preserving (respectively, surjective) cubic stochastic operators on the $2$ -dimensional simplex.
Mukhamedov, Farrukh; Fadillah Embong, Ahmad; Rosli, Azizi
In the present paper, we consider cubic stochastic operators, and prove that the surjectivity of such operators is equivalent to their orthogonal-preserving property. In the last section we provide a full description of orthogonal-preserving (respectively, surjective) cubic stochastic operators on the $2$ -dimensional simplex.
Hai, Guojun; Zhang, Nan
The aim of this article is to study the Fredholm completion problem of two-by-two partial operator matrices in which the lower-left entry is unspecified and others are specified. By using the methods of operator matrix representation and operator equation, we obtain necessity and sufficiency conditions for the partial operator matrices to have a Fredholm completion with the property that the lower-right entry of its Fredholm inverses is specified.
Hai, Guojun; Zhang, Nan
The aim of this article is to study the Fredholm completion problem of two-by-two partial operator matrices in which the lower-left entry is unspecified and others are specified. By using the methods of operator matrix representation and operator equation, we obtain necessity and sufficiency conditions for the partial operator matrices to have a Fredholm completion with the property that the lower-right entry of its Fredholm inverses is specified.