scholarly journals Structure of Iso-Symmetric Operators

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 256
Author(s):  
Bhagwati Prashad Duggal ◽  
In-Hyoun Kim
Keyword(s):  
Matrix Representation ◽  
Triangular Matrix ◽  
Space Operator ◽  
Left Multiplication ◽  
Direct Sum ◽  
Upper Triangular Matrix ◽  
Symmetric Operators ◽  
Bilateral Shift ◽  
Positive Integers ◽  
Power Bounded

For a Hilbert space operator T∈B(H), let LT and RT∈B(B(H)) denote, respectively, the operators of left multiplication and right multiplication by T. For positive integers m and n, let ▵T∗,Tm(I)=(LT∗RT−I)m(I) and δT∗,Tn(I)=(LT∗−RT)m(I). The operator T is said to be (m,n)-isosymmetric if ▵T∗,TmδT∗,Tn(I)=0. Power bounded (m,n)-isosymmetric operators T∈B(H) have an upper triangular matrix representation T=T1T30T2∈B(H1⊕H2) such that T1∈B(H1) is a C0.-operator which satisfies δT1∗,T1n(I|H1)=0 and T2∈B(H2) is a C1.-operator which satisfies AT2=(Vu⊕Vb)|H2A, A=limt→∞T2∗tT2t, Vu is a unitary and Vb is a bilateral shift. If, in particular, T is cohyponormal, then T is the direct sum of a unitary with a C00-contraction.

scholarly journals Structure of n-quasi left m-invertible and related classes of operators

2020 ◽  
Vol 53 (1) ◽  
pp. 249-268
Author(s):  
Bhagwati Prashad Duggal ◽  
In Hyun Kim
Keyword(s):  
Hilbert Space ◽  
Positive Integer ◽  
Direct Sum ◽  
Hilbert Space Operators ◽  
Isolated Points ◽  
Elementary Operators ◽  
Adjoint Operators ◽  
Number Of Classes ◽  
Symmetric Operators ◽  
Power Bounded

AbstractGiven Hilbert space operators T,S\in B( {\mathcal H} ), let \text{Δ} and \delta \in B(B( {\mathcal H} )) denote the elementary operators {\text{Δ}}_{T,S}(X)=({L}_{T}{R}_{S}-I)(X)=TXS-X and {\delta }_{T,S}(X)=({L}_{T}-{R}_{S})(X)=TX-XS. Let d=\text{Δ} or \delta . Assuming T commutes with {S}^{\ast }, and choosing X to be the positive operator {S}^{\ast n}{S}^{n} for some positive integer n, this paper exploits properties of elementary operators to study the structure of n-quasi {[}m,d]-operators {d}_{T,S}^{m}(X)=0 to bring together, and improve upon, extant results for a number of classes of operators, such as n-quasi left m-invertible operators, n-quasi m-isometric operators, n-quasi m-self-adjoint operators and n-quasi (m,C) symmetric operators (for some conjugation C of {\mathcal H} ). It is proved that {S}^{n} is the perturbation by a nilpotent of the direct sum of an operator {S}_{1}^{n}={\left(S{|}_{\overline{{S}^{n}( {\mathcal H} )}}\right)}^{n} satisfying {d}_{{T}_{1},{S}_{1}}^{m}({I}_{1})=0, {{T}_{1}=T}_{\overline{{S}^{n}( {\mathcal H} )}}, with the 0 operator; if S is also left invertible, then {S}^{n} is similar to an operator B such that {d}_{{B}^{\ast },B}^{m}(I)=0. For power bounded S and T such that S{T}^{\ast }-{T}^{\ast }S=0 and {\text{Δ}}_{T,S}({S}^{\ast n}{S}^{n})=0, S is polaroid (i.e., isolated points of the spectrum are poles). The product property, and the perturbation by a commuting nilpotent property, of operators T,S satisfying {d}_{T,S}^{m}(I)=0, given certain commutativity properties, transfers to operators satisfying {S}^{\ast n}{d}_{T,S}^{m}(I){S}^{n}=0.

scholarly journals Algebras of right ample semigroups

2018 ◽  
Vol 16 (1) ◽  
pp. 842-861 ◽  
Author(s):  
Junying Guo ◽  
Xiaojiang Guo
Keyword(s):  
Inverse Semigroup ◽  
Matrix Representation ◽  
Triangular Matrix ◽  
Inverse Semigroups ◽  
Generalized Matrix ◽  
Upper Triangular Matrix

AbstractStrict RA semigroups are common generalizations of ample semigroups and inverse semigroups. The aim of this paper is to study algebras of strict RA semigroups. It is proved that any algebra of strict RA semigroups with finite idempotents has a generalized matrix representation whose degree is equal to the number of non-zero regular 𝓓-classes. In particular, it is proved that any algebra of finite right ample semigroups has a generalized upper triangular matrix representation whose degree is equal to the number of non-zero regular 𝓓-classes. As its application, we determine when an algebra of strict RA semigroups (right ample monoids) is semiprimitive. Moreover, we prove that an algebra of strict RA semigroups (right ample monoids) is left self-injective iff it is right self-injective, iff it is Frobenius, and iff the semigroup is a finite inverse semigroup.

scholarly journals Accurate and Efficient Derivative-Free Three-Phase Power Flow Method for Unbalanced Distribution Networks

Computation ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 61
Author(s):  
Oscar Danilo Montoya ◽  
Juan S. Giraldo ◽  
Luis Fernando Grisales-Noreña ◽  
Harold R. Chamorro ◽  
Lazaro Alvarado-Barrios
Keyword(s):  
Power Flow ◽  
Triangular Matrix ◽  
Distribution Networks ◽  
Flow Method ◽  
Processing Times ◽  
Three Phase ◽  
Derivative Free ◽  
Upper Triangular Matrix ◽  
Power Flow Problem ◽  
Power Flow Method

The power flow problem in three-phase unbalanced distribution networks is addressed in this research using a derivative-free numerical method based on the upper-triangular matrix. The upper-triangular matrix is obtained from the topological connection among nodes of the network (i.e., through a graph-based method). The main advantage of the proposed three-phase power flow method is the possibility of working with single-, two-, and three-phase loads, including Δ- and Y-connections. The Banach fixed-point theorem for loads with Y-connection helps ensure the convergence of the upper-triangular power flow method based an impedance-like equivalent matrix. Numerical results in three-phase systems with 8, 25, and 37 nodes demonstrate the effectiveness and computational efficiency of the proposed three-phase power flow formulation compared to the classical three-phase backward/forward method and the implementation of the power flow problem in the DigSILENT software. Comparisons with the backward/forward method demonstrate that the proposed approach is 47.01%, 47.98%, and 36.96% faster in terms of processing times by employing the same number of iterations as when evaluated in the 8-, 25-, and 37-bus systems, respectively. An application of the Chu-Beasley genetic algorithm using a leader–follower optimization approach is applied to the phase-balancing problem utilizing the proposed power flow in the follower stage. Numerical results present optimal solutions with processing times lower than 5 s, which confirms its applicability in large-scale optimization problems employing embedding master–slave optimization structures.

scholarly journals Functional identities on upper triangular matrix rings

2020 ◽  
Vol 18 (1) ◽  
pp. 182-193
Author(s):  
He Yuan ◽  
Liangyun Chen
Keyword(s):  
Triangular Matrix ◽  
Matrix Rings ◽  
Unital Ring ◽  
Functional Identities ◽  
Upper Triangular Matrix ◽  
Free Subset

Abstract Let R be a subset of a unital ring Q such that 0 ∈ R. Let us fix an element t ∈ Q. If R is a (t; d)-free subset of Q, then Tn(R) is a (t′; d)-free subset of Tn(Q), where t′ ∈ Tn(Q), $\begin{array}{} t_{ll}' \end{array} $ = t, l = 1, 2, …, n, for any n ∈ N.

Piecewise Hereditary Triangular Matrix Algebras

2021 ◽  
Vol 28 (01) ◽  
pp. 143-154
Author(s):  
Yiyu Li ◽  
Ming Lu
Keyword(s):  
Positive Integer ◽  
Triangular Matrix ◽  
Derived Categories ◽  
Matrix Algebras ◽  
Tensor Algebras ◽  
Finite Dimensional ◽  
Abelian Categories ◽  
Upper Triangular Matrix ◽  
Orbit Categories ◽  
Finite Dimensional Algebras

For any positive integer [Formula: see text], we clearly describe all finite-dimensional algebras [Formula: see text] such that the upper triangular matrix algebras [Formula: see text] are piecewise hereditary. Consequently, we describe all finite-dimensional algebras [Formula: see text] such that their derived categories of [Formula: see text]-complexes are triangulated equivalent to derived categories of hereditary abelian categories, and we describe the tensor algebras [Formula: see text] for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.

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