scholarly journals Factorization of hyponormal operators

1972 ◽  
Vol 13 (3) ◽  
pp. 323-326
Author(s):  
Mary R. Embry
Keyword(s):  
Hilbert Space ◽  
Linear Operators ◽  
Continuous Linear ◽  
Hyponormal Operators ◽  
Image Position ◽  
Continuous Linear Operators

In [1]R. G. Douglas proved that if A and B are continuous linear operators on a Hilbert space X, the following three conditions are equivalent:

scholarly journals Periodicity of functionals and representations of normed algebras on reflexive spaces

1976 ◽  
Vol 20 (2) ◽  
pp. 99-120 ◽  
Author(s):  
N. J. Young
Keyword(s):  
Hilbert Space ◽  
Continuous Homomorphism ◽  
Linear Operators ◽  
The Other ◽  
Continuous Linear ◽  
Normed Algebra ◽  
Reflexive Space ◽  
To Come ◽  
Algebra Of Operators ◽  
Continuous Linear Operators

It is a well-known fact that any normed algebra can be represented isometrically as an algebra of operators with the operator norm. As might be expected from the very universality of this property, it is little used in the study of the structure of an algebra. Far more helpful are representations on Hilbert space, though these are correspondingly hard to come by: isometric representations on Hilbert space are not to be expected in general, and even continuous nontrivial representations may fail to exist. The purpose of this paper is to examine a class of representations intermediate in both availability and utility to those already mentioned—namely, representations on reflexive spaces. There certainly are normed algebras which admit isometric representations of the latter type but have not even faithful representations on Hilbert space: the most natural example is the algebra of all continuous linear operators on E where E = lp with 1 < p ≠ 2 < ∞, for Berkson and Porta proved in (2) that if E, F are taken from the spaces lp with 1 < p < ∞ and E ≠ F then the only continuous homomorphism from into is the zero mapping. On the other hand there are also algebras which have no continuous nontrivial representation on any reflexive space—for example the algebra of finite-rank operators on an irreflexive Banach space (see Berkson and Porta (2) or Barnes (1) or Theorem 3, Corollary 1 below).

Generalized Resolvent Equations and Unsymmetric Dirichlet Spaces

1981 ◽  
Vol 33 (5) ◽  
pp. 1111-1141
Author(s):  
Joanne Elliott
Keyword(s):  
Linear Operators ◽  
Dirichlet Spaces ◽  
Resolvent Equation ◽  
Continuous Linear ◽  
Generalized Resolvent ◽  
Resolvent Equations ◽  
Measure Spaces ◽  
Image Position ◽  
Generalized Resolvent Equations ◽  
Continuous Linear Operators

Let (X, , μ) and (X, , μ′) be measure spaces with the measures μ and μ′ totally finite. Suppose {Uλ: λ > 0} is a family of positive (i.e., ϕ ≧ 0 ⇒ Uλϕ ≧ 0) continuous linear operators from L2(X, dμ′) to L2(X,dμ) with the following additional properties: if ϕ ≧ 0 then Uλϕ is non-decreasing as λ increases, while λ−1Uλϕ is nonincreasing.A family {Mλ:λ > 0} of continuous linear operators from L2(X, dμ) to L2(X, dμ′) satisfies the “generalized resolvent equation” relative to {Uλ} if(0.1)for positive λ and v. If Uλ = λI, then (0.1) is just the well-known resolvent equation. The family {Mλ} is called submarkov if Mλ is a positive operator and(0.2)it is conservative if(0.3)

On eigenvalues and inverse singular values of compact linear operators in Hilbert space

1953 ◽  
Vol 49 (2) ◽  
pp. 201-212 ◽  
Author(s):  
J. P. O. Silberstein ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Continuous Linear Operator ◽  
Singular Values ◽  
Linear Operators ◽  
Complex Numbers ◽  
Identity Operator ◽  
Continuous Linear ◽  
Completely Continuous ◽  
Image Position

1·1. In this paper we shall be concerned with the equationswhere K is a compact (completely continuous) linear operator in a Hilbert space , K is the adjoint of K, I is the identity operator, x and y are elements of ∥ x ∥ denotes the norm of x, and κ and σ are complex numbers.

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.

scholarly journals Implicit Function Theorem. Part II

2019 ◽  
Vol 27 (2) ◽  
pp. 117-131
Author(s):  
Kazuhisa Nakasho ◽  
Yasunari Shidama
Keyword(s):  
Implicit Function Theorem ◽  
Existence And Uniqueness ◽  
Continuous Linear Operator ◽  
Linear Operators ◽  
Implicit Function ◽  
Continuous Linear ◽  
Lipschitz Continuous ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Continuous Linear Operators

Summary In this article, we formalize differentiability of implicit function theorem in the Mizar system [3], [1]. In the first half section, properties of Lipschitz continuous linear operators are discussed. Some norm properties of a direct sum decomposition of Lipschitz continuous linear operator are mentioned here. In the last half section, differentiability of implicit function in implicit function theorem is formalized. The existence and uniqueness of implicit function in [6] is cited. We referred to [10], [11], and [2] in the formalization.

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