On convergence to the Dirac measure and to the identity operator

Let Q a compact and convex subset of R('k), k (GREATERTHEQ) 1 and let {L(,j)}(,j(ELEM)(, )) be a sequence of positive linear operators from C('n)(Q), (n (ELEM) ('+)) to C(Q). The convergence of L(,j) to the identity operator I is closely related to the weak convergence of a sequence of finite measures (mu)(,j) to the unit (Dirac) measure (delta)(,x0), x(,0) (ELEM) Q.New estimates are given for the remainder (VBAR)(INT)(,Q)f d(mu)(,j) -- f(x(,0))(VBAR), where f (ELEM) C('n)(Q). Using moments methods, Shisha-Mond type best or nearly best upper-bounds are established for various choices of k, Q, n and given moments of (mu)(,j). Some of them lead to attainable inequalities. The optimal functions/measures are typically spline functions and finitely supported measures. The corresponding inequalities involve various measures of smoothness of f such as the first or second modulus of continuity of f('(n)), the Peetre K-functional of f or certain modifications and generalizations.Finally some miscellaneous sharp inequalities are obtained. These lead to Korovkin type convergence theorems relative to ratios of Fourier-Stieltjes hyperbolic coefficients.

Compact linear operators from an algebraic standpoint

Glasgow Mathematical Journal ◽

10.1017/s0017089500000069 ◽

1967 ◽

Vol 8 (1) ◽

pp. 41-49 ◽

Cited By ~ 4

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

Georgian Mathematical Journal ◽

10.1515/gmj.2010.039 ◽

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

Canadian Journal of Mathematics ◽

10.4153/cjm-1958-042-4 ◽

1958 ◽

Vol 10 ◽

pp. 431-446 ◽

Cited By ~ 13

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

Canadian Journal of Mathematics ◽

10.4153/cjm-1961-047-7 ◽

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

Canadian Journal of Mathematics ◽

10.4153/cjm-1985-033-2 ◽

1985 ◽

Vol 37 (4) ◽

pp. 635-643 ◽

Cited By ~ 2

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

Stochastics and Dynamics ◽

10.1142/s0219493721500155 ◽

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.

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

The Journal of Chemical Physics ◽

10.1063/1.5082596 ◽

2019 ◽

Vol 150 (7) ◽

pp. 071101 ◽

Cited By ~ 19

Author(s):

Maximilian A. C. Saller ◽

Aaron Kelly ◽

Jeremy O. Richardson

Keyword(s):

Identity Operator ◽

Semiclassical Dynamics

The Dirac Measure as Applied to the Solution of Difference Equations by Means of Transforms

American Mathematical Monthly ◽

10.1080/00029890.1959.11989333 ◽

1959 ◽

Vol 66 (6) ◽

pp. 483-485

Author(s):

Harry Lass ◽

Oldwig Von Roos

Keyword(s):

Difference Equations ◽

Dirac Measure

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

Glasgow Mathematical Journal ◽

10.1017/s0017089500002913 ◽

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.