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Discussions on partial isometries in Banach spaces and Banach algebras
Abdullah Alahmari,Mohamed Mabrouk,Mohamed Aziz Taoudi 대한수학회 2017 대한수학회보 Vol.54 No.2
The aim of this paper is twofold. Firstly, we introduce the concept of semi-partial isometry in a Banach algebra and carry out a comparison and a classification study for this concept. In particular, we show that in the context of $C^*$-algebras this concept coincides with the notion of partial isometry. Our results encompass several earlier ones concerning partial isometries in Hilbert spaces, Banach spaces and $C^*$-algebras. Finally, we study the notion of $(m, p)$-semi partial isometries.
DISCUSSIONS ON PARTIAL ISOMETRIES IN BANACH SPACES AND BANACH ALGEBRAS
Alahmari, Abdulla,Mabrouk, Mohamed,Taoudi, Mohamed Aziz Korean Mathematical Society 2017 대한수학회보 Vol.54 No.2
The aim of this paper is twofold. Firstly, we introduce the concept of semi-partial isometry in a Banach algebra and carry out a comparison and a classification study for this concept. In particular, we show that in the context of $C^*$-algebras this concept coincides with the notion of partial isometry. Our results encompass several earlier ones concerning partial isometries in Hilbert spaces, Banach spaces and $C^*$-algebras. Finally, we study the notion of (m, p)-semi partial isometries.
Sriwulan Adji,Saeid Zahmatkesh 대한수학회 2015 대한수학회지 Vol.52 No.4
Let Г + be the positive cone in a totally ordered abelian group Γ, and α an action of Г+ by extendible endomorphisms of a C∗-algebra A. Suppose I is an extendible α-invariant ideal of A. We prove that the partial-isometric crossed product I := I ×piso α Г+ embeds naturally as an ideal of A× piso α Г+, such that the quotient is the partial-isometric crossed product of the quotient algebra. We claim that this ideal I together with the kernel of a natural homomorphism Ø : A× piso α + → A × piso α Г+ gives a composition series of ideals of A ×piso α Г+ studied by Lindiarni and Raeburn.
ADJI, SRIWULAN,ZAHMATKESH, SAEID Korean Mathematical Society 2015 대한수학회지 Vol.52 No.4
Let ${\Gamma}^+$ be the positive cone in a totally ordered abelian group ${\Gamma}$, and ${\alpha}$ an action of ${\Gamma}^+$ by extendible endomorphisms of a $C^*$-algebra A. Suppose I is an extendible ${\alpha}$-invariant ideal of A. We prove that the partial-isometric crossed product $\mathcal{I}:=I{\times}^{piso}_{\alpha}{\Gamma}^+$ embeds naturally as an ideal of $A{\times}^{piso}_{\alpha}{\Gamma}^+$, such that the quotient is the partial-isometric crossed product of the quotient algebra. We claim that this ideal $\mathcal{I}$ together with the kernel of a natural homomorphism $\phi:A{\times}^{piso}_{\alpha}{\Gamma}^+{\rightarrow}A{\times}^{iso}_{\alpha}{\Gamma}^+$ gives a composition series of ideals of $A{\times}^{piso}_{\alpha}{\Gamma}^+$ studied by Lindiarni and Raeburn.
Fernandes, Vitor H.,Quinteiro, Teresa M. Korean Mathematical Society 2016 대한수학회보 Vol.53 No.2
In this note we consider the monoid $\mathcal{PODI}_n$ of all monotone partial permutations on $\{1,{\ldots},n\}$ and its submonoids $\mathcal{DP}_n$, $\mathcal{POI}_n$ and $\mathcal{ODP}_n$ of all partial isometries, of all order-preserving partial permutations and of all order-preserving partial isometries, respectively. We prove that both the monoids $\mathcal{POI}_n$ and $\mathcal{ODP}_n$ are quotients of bilateral semidirect products of two of their remarkable submonoids, namely of extensive and of co-extensive transformations. Moreover, we show that $\mathcal{PODI}_n$ is a quotient of a semidirect product of $\mathcal{POI}_n$ and the group $\mathcal{C}_2$ of order two and, analogously, $\mathcal{DP}_n$ is a quotient of a semidirect product of $\mathcal{ODP}_n$ and $\mathcal{C}_2$.
V\'\i tor H. Fernandes,Teresa M. Quinteiro 대한수학회 2016 대한수학회보 Vol.53 No.2
In this note we consider the monoid $\PODI_n$ of all monotone partial permutations on $\{1,\ldots,n\}$ and its submonoids $\DP_n$, $\POI_n$ and $\ODP_n$ of all partial isometries, of all order-preserving partial permutations and of all order-preserving partial isometries, respectively. We prove that both the monoids $\POI_n$ and $\ODP_n$ are quotients of bilateral semidirect products of two of their remarkable submonoids, namely of extensive and of co-extensive transformations. Moreover, we show that $\PODI_n$ is a quotient of a semidirect product of $\POI_n$ and the group $\mathcal{C}_2$ of order two and, analogously, $\DP_n$ is a quotient of a semidirect product of $\ODP_n$ and $\mathcal{C}_2$.
Weak normal properties of partial isometries
Ting Liu,Yanying Men,Sen Zhu 대한수학회 2019 대한수학회지 Vol.56 No.6
This paper describes when a partial isometry satisfies several weak normal properties. Topics treated include quasi-normality, subnormality, hyponormality, $p$-hyponormality ($p>0)$, $w$-hyponormality, paranormality, normaloidity, spectraloidity, the von Neumann property and Weyl's theorem.
WEAK NORMAL PROPERTIES OF PARTIAL ISOMETRIES
Liu, Ting,Men, Yanying,Zhu, Sen Korean Mathematical Society 2019 대한수학회지 Vol.56 No.6
This paper describes when a partial isometry satisfies several weak normal properties. Topics treated include quasi-normality, subnormality, hyponormality, p-hyponormality (p > 0), w-hyponormality, paranormality, normaloidity, spectraloidity, the von Neumann property and Weyl's theorem.
Space-time fractional stochastic partial differential equations
Mijena, J.B.,Nane, E. North-Holland Pub. Co ; Elsevier Science Ltd 2015 Stochastic processes and their applications Vol.125 No.9
We consider non-linear time-fractional stochastic heat type equation @?<SUB>t</SUB><SUP>β</SUP>u<SUB>t</SUB>(x)=-ν(-Δ)<SUP>α/2</SUP>u<SUB>t</SUB>(x)+I<SUB>t</SUB><SUP>1-β</SUP>[σ(u)W@?(t,x)] in (d+1) dimensions, where ν>0,β@?(0,1), α@?(0,2] and d<min{2,β<SUP>-1</SUP>}α, @?<SUB>t</SUB><SUP>β</SUP> is the Caputo fractional derivative, -(-Δ)<SUP>α/2</SUP> is the generator of an isotropic stable process, I<SUB>t</SUB><SUP>1-β</SUP> is the fractional integral operator, W@?(t,x) is space-time white noise, and σ:R→R is Lipschitz continuous. Time fractional stochastic heat type equations might be used to model phenomenon with random effects with thermal memory. We prove existence and uniqueness of mild solutions to this equation and establish conditions under which the solution is continuous. Our results extend the results in the case of parabolic stochastic partial differential equations obtained in Foondun and Khoshnevisan (2009), Walsh (1986). In sharp contrast to the stochastic partial differential equations studied earlier in Foondun and Khoshnevisan (2009), Khoshnevisan (2014) and Walsh (1986), in some cases our results give existence of random field solutions in spatial dimensions d=1,2,3. Under faster than linear growth of σ, we show that time fractional stochastic partial differential equation has no finite energy solution. This extends the result of Foondun and Parshad (in press) in the case of parabolic stochastic partial differential equations. We also establish a connection of the time fractional stochastic partial differential equations to higher order parabolic stochastic differential equations.
Generalized Inverses and Solutions to Equations in Rings with Involution
Yue Sui,Junchao Wei 경북대학교 자연과학대학 수학과 2024 Kyungpook mathematical journal Vol.64 No.1
In this paper, we focus on partial isometry elements and strongly EP elements or a ring. We construct characterizing equations such that an element which both group invertible and MP-invertible, is a partial isometry element, or is strongly EP, exactly when these equations have solution in a given set.