scholarly journals A formula to calculate the spectral radius of a compact linear operator

1997 ◽  
Vol 20 (3) ◽  
pp. 585-588 ◽  
Author(s):  
Fernando Garibay Bonales ◽  
Rigoberto Vera Mendoza
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Vector Space ◽  
Compact Operator ◽  
Spectral Radius ◽  
Topological Vector Space ◽  
Closed Subspace ◽  
Minkowski Functional ◽  
Normed Spaces ◽  
Locally Convex

There is a formula (Gelfand's formula) to find the spectral radius of a linear operator defined on a Banach space. That formula does not apply even in normed spaces which are not complete. In this paper we show a formula to find the spectral radius of any linear and compact operatorTdefined on a complete topological vector space, locally convex. We also show an easy way to find a non-trivialT-invariant closed subspace in terms of Minkowski functional.

scholarly journals Linear operators whose domain is locally convex

1977 ◽  
Vol 20 (4) ◽  
pp. 293-299 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Vector Space ◽  
Unit Ball ◽  
Topological Vector Space ◽  
Continuous Linear Operator ◽  
Linear Operators ◽  
Continuous Linear ◽  
Locally Convex

Let F be an arbitrary topological vector space; we shall say that a subset S of F is quasi-convex if the set of continuous affine functionals on S separates the points of S. If X is a Banach space and T : X → F is a continuous linear operator, then T is quasi-convex if is quasi-convex, where U is the unit ball of X.

scholarly journals Finitely spectral operators

1986 ◽  
Vol 28 (1) ◽  
pp. 95-112 ◽  
Author(s):  
B. Nagy
Keyword(s):  
Banach Space ◽  
Vector Space ◽  
Topological Vector Space ◽  
General Situation ◽  
Countable Additivity ◽  
Resolution Of The Identity ◽  
Set Function ◽  
Locally Convex

In the theory of spectral (and prespectral) operators in a Banach space or in a locally convex topological vector space the countable additivity (in some topology) of a resolution of the identity of the operator is a standing assumption. One might wonder why. Even if one cannot completely agree with the opinion of Diestel and Uhl ([6, p. 32]) stating that “countable additivity [of a set function] is often more of a hindrance than a help”, it might be interesting to study which portions of the theory of (pre)spectral operators and in which form extend to the more general situation described below.

Schauder bases and decompositions in locally convex spaces

1974 ◽  
Vol 76 (1) ◽  
pp. 145-152 ◽  
Author(s):  
J. H. Webb
Keyword(s):  
Linear Operator ◽  
Vector Space ◽  
Topological Vector Space ◽  
Weak Topology ◽  
Locally Convex Spaces ◽  
Partial Summation ◽  
Schauder Bases ◽  
Summation Operator ◽  
Image Position ◽  
Locally Convex

Let E[τ] be a locally convex Hausdorif topological vector space, with a Schauder basis {xi, x′j wherefor each x ∈ E. The partial summation operator Sn, defined byis a linear operator on E, whose definition extends at once to a linear operator mapping (E′)* into E, where (E′)* is the algebraic dual of E′. The dual of Sn is the operator S′n, mapping E* into E′, defined byand 〈Snx, x′〉 = 〈x, S′nx′〉 for each x ∈ (E′)*. It is easy to see that S′nx′ → x′ with respect to the weak topology σ(E′, E) for each x′ ∈ E′.

Banach Envelopes of Non-Locally Convex Spaces

1986 ◽  
Vol 38 (1) ◽  
pp. 65-86 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Convex Hull ◽  
Bounded Linear Operator ◽  
Locally Convex Spaces ◽  
Minkowski Functional ◽  
Bounded Linear ◽  
Banach Envelope ◽  
Image Position ◽  
Locally Convex

Let X be a quasi-Banach space whose dual X* separates the points of X. Then X* is a Banach space under the normFrom X we can construct the Banach envelope Xc of X by defining for x ∊ X, the normThen Xc is the completion of (X, ‖ ‖c). Alternatively ‖ ‖c is the Minkowski functional of the convex hull of the unit ball. Xc has the property that any bounded linear operator L:X → Z into a Banach space extends with preservation of norm to an operator .

On the Dimension of a Complete Metrizable Topological Vector Space

1977 ◽  
Vol 20 (2) ◽  
pp. 271-272 ◽  
Author(s):  
J. O. Popoola ◽  
I. Tweddle
Keyword(s):  
Banach Space ◽  
Vector Space ◽  
Banach Spaces ◽  
Topological Vector Space ◽  
Fréchet Spaces ◽  
Convex Case ◽  
Space Case ◽  
Locally Convex ◽  
Elegant Proof ◽  
Metrizable Topological Vector Space

The purpose of this note is to prove a result which is known to hold for Fréchet spaces [1, Chapitre II, §5, Exercise 24]. M. M. Day [2, p. 37] attributes the Banach space case to H. Löwig, although the earliest version that we have been able to find is that given by G. W. Mackey in [7, Theorem 1-1]. Recently H. E. Lacey has given an elegant proof for Banach spaces [5]. It is perhaps interesting to note that the non-locally convex case can be deduced from these known results which are established by duality arguments.

scholarly journals Superdiagonal forms for completely continuous linear operators

1982 ◽  
Vol 23 (2) ◽  
pp. 163-170 ◽  
Author(s):  
Demetrios Koros
Keyword(s):  
Linear Operator ◽  
Vector Space ◽  
Topological Vector Space ◽  
Continuous Linear Operator ◽  
Linear Operators ◽  
Complex Field ◽  
Continuous Linear ◽  
Completely Continuous ◽  
Hausdorff Topological Vector Space ◽  
Locally Convex

Altman [1] showed that Riesz-Schauder theory remains valid for a completely continuous linear operator on a locally convex Hausdorflf topological vector space over the complex field. In a later paper [2], he proved an analogue of the Aronszajn-Smith result; specifically, he showed that such an operator possesses a proper closed invariant subspace. The purpose of this paper is to show that Ringrose's theory of superdiagonal forms for compact linear operators [3] can be generalized to the case of a completely continuous linear operator on a locally convex Hausdorff topological vector space over the complex field. However, the proof given in [3] requires considerable modification.

scholarly journals In what spaces is every closed normal cone regular?

1970 ◽  
Vol 17 (2) ◽  
pp. 121-125 ◽  
Author(s):  
C. W. McArthur
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Unconditional Basis ◽  
Normal Cone ◽  
Closed Subspace ◽  
Locally Convex Space ◽  
Regular Part ◽  
Fréchet Space ◽  
Sequential Completeness ◽  
Locally Convex

It is known (13, p. 92) that each closed normal cone in a weakly sequentially complete locally convex space is regular and fully regular. Part of the main theorem of this paper shows that a certain amount of weak sequential completeness is necessary in order that each closed normal cone be regular. Specifically, it is shown that each closed normal cone in a Fréchet space is regular if and only if each closed subspace with an unconditional basis is weakly sequentially complete. If E is a strongly separable conjugate of a Banach space it is shown that each closed normal cone in E is fully regular. If E is a Banach space with an unconditional basis it is shown that each closed normal cone in E is fully regular if and only if E is the conjugate of a Banach space.

scholarly journals A class of factorable topological algebras

1990 ◽  
Vol 33 (1) ◽  
pp. 53-59 ◽  
Author(s):  
E. Ansari-Piri
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Approximate Identity ◽  
Necessary Condition ◽  
Topological Algebras ◽  
Approximate Identities ◽  
Cohen Factorization ◽  
Locally Convex ◽  
Space Property ◽  
Uniformly Bounded

The famous Cohen factorization theorem, which says that every Banach algebra with bounded approximate identity factors, has already been generalized to locally convex algebras with what may be termed “uniformly bounded approximate identities”. Here we introduce a new notion, that of fundamentality generalizing both local boundedness and local convexity, and we show that a fundamental Fréchet algebra with uniformly bounded approximate identity factors. Fundamentality is a topological vector space property rather than an algebra property. We exhibit some non-fundamental topological vector space and give a necessary condition for Orlicz space to be fundamental.

scholarly journals Almost Surjective Epsilon-Isometry in The Reflexive Banach Spaces

CAUCHY ◽  
2017 ◽  
Vol 4 (4) ◽  
pp. 167
Author(s):  
Minanur Rohman
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Lorentz Space ◽  
Bounded Linear Operator ◽  
Closed Subspace ◽  
Reflexive Banach Space ◽  
Reflexive Banach Spaces ◽  
Bounded Linear ◽  
Star Topology

<p class="AbstractCxSpFirst">In this paper, we will discuss some applications of almost surjective epsilon-isometry mapping, one of them is in Lorentz space ( L_(p,q)-space). Furthermore, using some classical theorems of w star-topology and concept of closed subspace -complemented, for every almost surjective epsilon-isometry mapping  <em>f </em>: <em>X to</em><em> Y</em>, where <em>Y</em> is a reflexive Banach space, then there exists a bounded linear operator   <em>T</em> : <em>Y to</em><em> X</em>  with  such that</p><p class="AbstractCxSpMiddle">  </p><p class="AbstractCxSpLast">for every x in X.</p>

scholarly journals Asymptotic Farkas lemmas for convex systems

2016 ◽  
Vol 19 (4) ◽  
pp. 160-168
Author(s):  
Dinh Nguyen ◽  
Mo Hong Tran
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Constraint Qualification ◽  
Convex Set ◽  
Lower Semicontinuous ◽  
Closed Convex Cone ◽  
Convex Mapping ◽  
Hausdorff Topological Vector Space ◽  
Reverse Convex ◽  
Locally Convex

In this paper we establish characterizations of the containment of the set {xX: xC,g(x)K}{xX: f (x)0}, where C is a closed convex subset of a locally convex Hausdorff topological vector space, X, K is a closed convex cone in another locally convex Hausdorff topological vector space and g:X Y is a K- convex mapping, in a reverse convex set, define by the proper, lower semicontinuous, convex function. Here, no constraint qualification condition or qualification condition are assumed. The characterizations are often called asymptotic Farkas-type results. The second part of the paper was devoted to variant Asymptotic Farkas-type results where the mapping is a convex mapping with respect to an extended sublinear function. It is also shown that under some closedness conditions, these asymptotic Farkas lemmas go back to non-asymptotic Farkas lemmas or stable Farkas lemmas established recently in the literature. The results can be used to study the optimization

Sign in / Sign up

Export Citation Format

Share Document