On the decomposition and integral representation of continuous linear operators

1961 ◽  
Vol 53 (1) ◽  
pp. 63-87 ◽  
Author(s):  
Alexandra Ionescu Tulcea ◽  
Cassius Ionescu Tulcea
2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Marian Nowak

Let (Ω,Σ,μ) be a complete σ-finite measure space, φ be a Young function, and X and Y be Banach spaces. Let Lφ(X) denote the Orlicz-Bochner space, and Tφ∧ denote the finest Lebesgue topology on Lφ(X). We study the problem of integral representation of (Tφ∧,·Y)-continuous linear operators T:Lφ(X)→Y with respect to the representing operator-valued measures. The relationships between (Tφ∧,·Y)-continuous linear operators T:Lφ(X)→Y and the topological properties of their representing operator measures are established.


2010 ◽  
Vol 17 (4) ◽  
pp. 787-794
Author(s):  
Vaja Tarieladze

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.


1976 ◽  
Vol 20 (2) ◽  
pp. 99-120 ◽  
Author(s):  
N. J. Young

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).


2019 ◽  
Vol 27 (2) ◽  
pp. 117-131
Author(s):  
Kazuhisa Nakasho ◽  
Yasunari Shidama

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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