𝑚th roots of the identity operator and the geometry conjecture

2021 ◽  
Author(s):  
Stephen Simons
Keyword(s):  
Identity Operator

scholarly journals Compact linear operators from an algebraic standpoint

1967 ◽  
Vol 8 (1) ◽  
pp. 41-49 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Linear Operators ◽  
Identity Operator ◽  
Algebra Structure ◽  
Natural Setting ◽  
Norm Topology ◽  
Bounded Linear ◽  
Underlying Space ◽  
Closed Subalgebra

Let B(X) denote the Banach algebra of all bounded linear operators on a Banach space X. Let t be an element of B(X), and let edenote the identity operator on X. Since the earliest days of the theory of Banach algebras, ithas been understood that the natural setting within which to study spectral properties of t is the Banach algebra B(X), or perhaps a closed subalgebra of B(X) containing t and e. The effective application of this method to a given class of operators depends upon first translating the data into terms involving only the Banach algebra structure of B(X) without reference to the underlying space X. In particular, the appropriate topology is the norm topology in B(X) given by the usual operator norm. Theorem 1 carries out this translation for the class of compact operators t. It is proved that if t is compact, then multiplication by t is a compact linear operator on the closed subalgebra of B(X) consisting of operators that commute with t.

An operator version of Abel's continuity theorem

2010 ◽  
Vol 17 (4) ◽  
pp. 787-794
Author(s):  
Vaja Tarieladze
Keyword(s):  
Banach Space ◽  
Linear Operators ◽  
Identity Operator ◽  
Continuous Linear ◽  
Continuity Theorem ◽  
Operator Version ◽  
Continuous Linear Operators

Abstract For a Banach space X let 𝔄 be the set of continuous linear operators A : X → X with ‖A‖ < 1, I be the identity operator and 𝔄 c ≔ {A ∈ 𝔄 : ‖I – A‖ ≤ c(1 – ‖A‖)}, where c ≥ 1 is a constant. Let, moreover, (xk ) k≥0 be a sequence in X such that the series converges and ƒ : 𝔄 ∪ {I} → X be the mapping defined by the equality It is shown that ƒ is continuous on 𝔄 and for every c ≥ 1 the restriction of ƒ to 𝔄 c ∪ {I} is continuous at I.

Spectral Theory for the Differential Equation Lu= λ Mu

1958 ◽  
Vol 10 ◽  
pp. 431-446 ◽  
Author(s):  
Fred Brauer
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Spectral Theory ◽  
Real Line ◽  
Differential Operators ◽  
Identity Operator ◽  
The Real ◽  
Ordinary Differential Operators ◽  
Extensive Bibliography

Let L and M be linear ordinary differential operators defined on an interval I, not necessarily bounded, of the real line. We wish to consider the expansion of arbitrary functions in eigenfunctions of the differential equation Lu = λMu on I. The case where M is the identity operator and L has a self-adjoint realization as an operator in the Hilbert space L 2(I) has been treated in various ways by several authors; an extensive bibliography may be found in (4) or (8).

A Note on Induced Modules

1961 ◽  
Vol 13 ◽  
pp. 587-592
Author(s):  
Charles W. Curtis
Keyword(s):  
Identity Element ◽  
Identity Operator ◽  
Finitely Generated ◽  
Standard Construction ◽  
Left And Right

In this paper, A denotes a ring with an identity element 1, and B a subring of A containing 1 such that B satisfies the left and right minimum conditions, and A is a finitely generated left and right B-module. The identity element 1 is required to act as the identity operator on all modules which we shall consider. For any left B-module V, there is a standard construction of a left A -module which is, roughly speaking, the smallest A -module containing V.

Groups with Finite Dimensional Irreducible Multiplier Representations

1985 ◽  
Vol 37 (4) ◽  
pp. 635-643 ◽  
Author(s):  
A. K. Holzherr
Keyword(s):  
Von Neumann Algebra ◽  
Regular Representation ◽  
Sufficient Conditions ◽  
Locally Compact Group ◽  
Identity Operator ◽  
Central Projection ◽  
List Type ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Necessary And Sufficient

Let G be a locally compact group and ω a normalized multiplier on G. Denote by V(G) (respectively by V(G, ω)) the von Neumann algebra generated by the regular representation (respectively co-regular representation) of G. Kaniuth [6] and Taylor [14] have characterized those G for which the maximal type I finite central projection in V(G) is non-zero (respectively the identity operator in V(G)).In this paper we determine necessary and sufficient conditions on G and ω such that the maximal type / finite central projection in V(G, ω) is non-zero (respectively the identity operator in V(G, ω)) and construct this projection explicitly as a convolution operator on L2(G). As a consequence we prove the following statements are equivalent,(i) V(G, ω) is type I finite,(ii) all irreducible multiplier representations of G are finite dimensional,(iii) Gω (the central extension of G) is a Moore group, that is all its irreducible (ordinary) representations are finite dimensional.

Quantum Feller semigroup in terms of quantum Bernoulli noises

2020 ◽  
pp. 2150015
Author(s):  
Jinshu Chen
Keyword(s):  
Local Structure ◽  
Identity Operator ◽  
Sufficient Condition ◽  
Markov Semigroups ◽  
Feller Semigroup ◽  
Abelian Subalgebra ◽  
Feller Property ◽  
The Family ◽  
Invariant States ◽  
Creation Operators

Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal-time. In this paper, we aim to investigate quantum Feller semigroups in terms of QBN. We first investigate local structure of the algebra generated by identity operator and QBN. We then use our new results obtained here to construct a class of quantum Markov semigroups from QBN which enjoy Feller property. As an application of our results, we examine a special quantum Feller semigroup associated with QBN, when it reduced to a certain Abelian subalgebra, the semigroup gives rise to the semigroup generated by Ornstein–Uhlenbeck operator. Finally, we find a sufficient condition for the existence of faithful invariant states that are diagonal for the semigroup.

scholarly journals On the identity of the identity operator in nonadiabatic linearized semiclassical dynamics

2019 ◽  
Vol 150 (7) ◽  
pp. 071101 ◽  
Author(s):  
Maximilian A. C. Saller ◽  
Aaron Kelly ◽  
Jeremy O. Richardson
Keyword(s):  
Identity Operator ◽  
Semiclassical Dynamics

scholarly journals States which have a trace-like property relative to a C*-subalgebra of B(H)

1976 ◽  
Vol 17 (2) ◽  
pp. 158-160
Author(s):  
Guyan Robertson
Keyword(s):  
Hilbert Space ◽  
Spectral Radius ◽  
Operator Theory ◽  
Linear Operators ◽  
Operator Norm ◽  
Identity Operator ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Image Position

In what follows, B(H) will denote the C*-algebra of all bounded linear operators on a Hilbert space H. Suppose we are given a C*-subalgebra A of B(H), which we shall suppose contains the identity operator 1. We are concerned with the existence of states f of B(H) which satisfy the following trace-like relation relative to A:Our first result shows the existence of states f satisfying (*), when A is the C*-algebra C*(x) generated by a normaloid operator × and the identity. This allows us to give simple proofs of some well-known results in operator theory. Recall that an operator × is normaloid if its operator norm equals its spectral radius.

scholarly journals Toeplitz Type Operators Associated with Generalized Calderón-Zygmund Operator on Weighted Morrey Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Bijun Ren ◽  
Enbin Zhang
Keyword(s):  
Morrey Space ◽  
Morrey Spaces ◽  
Linear Operators ◽  
Identity Operator ◽  
Zygmund Operator ◽  
Weighted Bmo ◽  
Weighted Morrey Space ◽  
Weighted Morrey Spaces ◽  
Bmo Spaces ◽  
Toeplitz Type Operator

LetT1be a generalized Calderón-Zygmund operator or±I(the identity operator), letT2andT4be the linear operators, and letT3=±I. Denote the Toeplitz type operator byTb=T1MbIαT2+T3IαMbT4, whereMbf=bfandIαis the fractional integral operator. In this paper, we investigate the boundedness of the operatorTbon weighted Morrey space whenbbelongs to the weighted BMO spaces.

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