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.

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.

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 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 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 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

scholarly journals Mean value theorems and a Taylor theorem for vector valued functions

1981 ◽  
Vol 24 (1) ◽  
pp. 69-77
Author(s):  
Rudolf Výborný
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Extension Theorem ◽  
Mean Value ◽  
Mean Value Theorems ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Convex

Two mean value theorems and a Taylor theorem for functions with values in a locally convex topological vector space are proved without the use of the Hahn-Banach extension theorem.

Directionally Lipschitzian Mappings on Baire Spaces

1984 ◽  
Vol 36 (1) ◽  
pp. 95-130 ◽  
Author(s):  
J. M. Borwein ◽  
H. M. Stròjwas
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Optimization Problems ◽  
Directional Derivatives ◽  
Primary Interest ◽  
Lipschitzian Functions ◽  
Lipschitzian Mappings ◽  
Generalized Directional Derivatives ◽  
Derivatives Of ◽  
Locally Convex

Studies of optimization problems have led in recent years to definitions of several types of generalized directional derivatives. Those derivatives of primary interest in this paper were introduced and investigated by F. M. Clarke ([5], [6], [7], [8]), J. B. Hiriart-Urruty ([12]), Lebourg ([16], [17]), R. T. Rockafellar ([23], [24], [26], [27]), Penot ([21], [22]) among others.In an attempt to explore in more detail relationships between various types of generalized directional derivatives we discovered some unexpected results which were not observed by the above mentioned authors. We are able to give simple conditions which characterize directionally Lipschitzian functions defined on a Baire metrizable locally convex topological vector space.

scholarly journals A few remarks on bounded operators on topological vector spaces

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 763-772
Author(s):  
Omid Zabeti ◽  
Ljubisa Kocinac
Keyword(s):  
Tensor Product ◽  
Vector Space ◽  
Topological Vector Space ◽  
Compact Operators ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Bounded Operators ◽  
Topological Vector Spaces ◽  
Different Types ◽  
Locally Convex

We give a few observations on different types of bounded operators on a topological vector space X and their relations with compact operators on X. In particular, we investigate when these bounded operators coincide with compact operators. We also consider similar types of bounded bilinear mappings between topological vector spaces. Some properties of tensor product operators between locally convex spaces are established. In the last part of the paper we deal with operators on topological Riesz spaces.

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