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Rakbud, Jittisak Department of Mathematics 2018 Kyungpook mathematical journal Vol.58 No.4
In this paper, we use the notion of characters of transformations provided in [8] by Purisang and Rakbud to define a notion of weak regularity of transformations on an arbitrarily fixed set X. The regularity of a semigroup of weakly regular transformations on a set X is also investigated.
SEQUENCE SPACES OF OPERATORS ON l<sub>2</sub>
Rakbud, Jitti,Ong, Sing-Cheong Korean Mathematical Society 2011 대한수학회지 Vol.48 No.6
In this paper, we define some new sequence spaces of infinite matrices regarded as operators on $l_2$ by using algebraic properties of such the matrices under the Schur product multiplication. Some of their basic properties as well as duality and preduality are discussed.
SCHATTEN'S THEOREM ON ABSOLUTE SCHUR ALGEBRAS
Rakbud, Jitti,Chaisuriya, Pachara Korean Mathematical Society 2008 대한수학회지 Vol.45 No.2
In this paper, we study duality in the absolute Schur algebras that were first introduced in [1] and extended in [5]. This is done in a way analogous to the classical Schatten's Theorem on the Banach space $B(l_2)$ of bounded linear operators on $l_2$ involving the duality relation among the class of compact operators K, the trace class $C_1$ and $B(l_2)$. We also study the reflexivity in such the algebras.
CONTINUITY OF BANACH ALGEBRA VALUED FUNCTIONS
Rakbud, Jittisak Korean Mathematical Society 2014 대한수학회논문집 Vol.29 No.4
Let K be a compact Hausdorff space, $\mathfrak{A}$ a commutative complex Banach algebra with identity and $\mathfrak{C}(\mathfrak{A})$ the set of characters of $\mathfrak{A}$. In this note, we study the class of functions $f:K{\rightarrow}\mathfrak{A}$ such that ${\Omega}_{\mathfrak{A}}{\circ}f$ is continuous, where ${\Omega}_{\mathfrak{A}}$ denotes the Gelfand representation of $\mathfrak{A}$. The inclusion relations between this class, the class of continuous functions, the class of bounded functions and the class of weakly continuous functions relative to the weak topology ${\sigma}(\mathfrak{A},\mathfrak{C}(\mathfrak{A}))$, are discussed. We also provide some results on its completeness under the norm defined by ${\mid}{\parallel}f{\parallel}{\mid}={\parallel}{\Omega}_{\mathfrak{A}}{\circ}f{\parallel}_{\infty}$.
SCHATTEN CLASSES OF MATRICES IN A GENERALIZED B(l<sub>2</sub>)
Rakbud, Jitti,Chaisuriya, Pachara Korean Mathematical Society 2010 대한수학회지 Vol.47 No.1
In this paper, we study a generalization of the Banach space B($l_2$) of all bounded linear operators on $l_2$. Over this space, we present some reasonable ways to define Schatten-type classes which are generalizations of the classical Schatten classes of compact operators on $l_2$.
SCHATTEN CLASSES OF MATRICES IN A GENERALIZED B(l2)
Jitti Rakbud,Pachara Chaisuriya 대한수학회 2010 대한수학회지 Vol.47 No.1
In this paper, we study a generalization of the Banach space B(l2) of all bounded linear operators on l2. Over this space, we present some reasonable ways to define Schatten-type classes which are generalizations of the classical Schatten classes of compact operators on l2.
Schatten's theorem on absolute Schur algebras
Jitti Rakbud,Pachara Chaisuriya 대한수학회 2008 대한수학회지 Vol.45 No.2
In this paper, we study duality in the absolute Schur algebras that were first introduced in [1] and extended in [5]. This is done in a way analogous to the classical Schatten’s Theorem on the Banach space B(l₂) of bounded linear operators on l₂ involving the duality relation among the class of compact operators K, the trace class C₁ and B(l₂). We also study the reflexivity in such the algebras. In this paper, we study duality in the absolute Schur algebras that were first introduced in [1] and extended in [5]. This is done in a way analogous to the classical Schatten’s Theorem on the Banach space B(l₂) of bounded linear operators on l₂ involving the duality relation among the class of compact operators K, the trace class C₁ and B(l₂). We also study the reflexivity in such the algebras.
SEQUENCE SPACES OF OPERATORS ON l_2
Jitti Rakbud,Sing-Cheong Ong 대한수학회 2011 대한수학회지 Vol.48 No.6
In this paper, we define some new sequence spaces of infinite matrices regarded as operators on l_2 by using algebraic properties of such the matrices under the Schur product multiplication. Some of their basic properties as well as duality and preduality are discussed.
REGULARITY OF TRANSFORMATION SEMIGROUPS DEFINED BY A PARTITION
Purisang, Pattama,Rakbud, Jittisak Korean Mathematical Society 2016 대한수학회논문집 Vol.31 No.2
Let X be a nonempty set, and let $\mathfrak{F}=\{Y_i:i{\in}I\}$ be a family of nonempty subsets of X with the properties that $X={\bigcup}_{i{\in}I}Y_i$, and $Y_i{\cap}Y_j={\emptyset}$ for all $i,j{\in}I$ with $i{\neq}j$. Let ${\emptyset}{\neq}J{\subseteq}I$, and let $T^{(J)}_{\mathfrak{F}}(X)=\{{\alpha}{\in}T(X):{\forall}i{\in}I{\exists}_j{\in}J,Y_i{\alpha}{\subseteq}Y_j\}$. Then $T^{(J)}_{\mathfrak{F}}(X)$ is a subsemigroup of the semigroup $T(X,Y^{(J)})$ of functions on X having ranges contained in $Y^{(J)}$, where $Y^{(J)}:={\bigcup}_{i{\in}J}Y_i$. For each ${\alpha}{\in}T^{(J)}_{\mathfrak{F}}(X)$, let ${\chi}^{({\alpha})}:I{\rightarrow}J$ be defined by $i{\chi}^{({\alpha})}=j{\Leftrightarrow}Y_i{\alpha}{\subseteq}Y_j$. Next, we define two congruence relations ${\chi}$ and $\widetilde{\chi}$ on $T^{(J)}_{\mathfrak{F}}(X)$ as follows: $({\alpha},{\beta}){\in}{\chi}{\Leftrightarrow}{\chi}^{({\alpha})}={\chi}^{({\beta})}$ and $({\alpha},{\beta}){\in}\widetilde{\chi}{\Leftrightarrow}{\chi}^{({\alpha})}{\mid}_J={\chi}^{({\alpha})}{\mid}_J$. We begin this paper by studying the regularity of the quotient semigroups $T^{(J)}_{\mathfrak{F}}(X)/{\chi}$ and $T^{(J)}_{\mathfrak{F}}(X)/{\widetilde{\chi}}$, and the semigroup $T^{(J)}_{\mathfrak{F}}(X)$. For each ${\alpha}{\in}T_{\mathfrak{F}}(X):=T^{(I)}_{\mathfrak{F}}(X)$, we see that the equivalence class [${\alpha}$] of ${\alpha}$ under ${\chi}$ is a subsemigroup of $T_{\mathfrak{F}}(X)$ if and only if ${\chi}^{({\alpha})}$ is an idempotent element in the full transformation semigroup T(I). Let $I_{\mathfrak{F}}(X)$, $S_{\mathfrak{F}}(X)$ and $B_{\mathfrak{F}}(X)$ be the sets of functions in $T_{\mathfrak{F}}(X)$ such that ${\chi}^{({\alpha})}$ is injective, surjective and bijective respectively. We end this paper by investigating the regularity of the subsemigroups [${\alpha}$], $I_{\mathfrak{F}}(X)$, $S_{\mathfrak{F}}(X)$ and $B_{\mathfrak{F}}(X)$ of $T_{\mathfrak{F}}(X)$.
MATRIX OPERATORS ON FUNCTION-VALUED FUNCTION SPACES
Ong, Sing-Cheong,Rakbud, Jitti,Wootijirattikal, Titarii The Kangwon-Kyungki Mathematical Society 2019 한국수학논문집 Vol.27 No.2
We study spaces of continuous-function-valued functions that have the property that composition with evaluation functionals induce $weak^*$ to norm continuous maps to ${\ell}^p$ space ($p{\in}(1,\;{\infty})$). Versions of $H{\ddot{o}}lder^{\prime}s$ inequality and Riesz representation theorem are proved to hold on these spaces. We prove a version of Dixmier's theorem for spaces of function-valued matrix operators on these spaces, and an analogue of the trace formula for operators on Hilbert spaces. When the function space is taken to be the complex field, the spaces are just the ${\ell}^p$ spaces and the well-known classical theorems follow from our results.