scholarly journals ON THE EXISTENCE OF NON-NORM-ATTAINING OPERATORS

2021 ◽  
pp. 1-13
Author(s):  
Sheldon Dantas ◽  
Mingu Jung ◽  
Gonzalo Martínez-Cervantes
Keyword(s):  
Sufficient Conditions ◽  
Compact Set ◽  
Closed Convex Hull ◽  
List Type ◽  
Weak Operator Topology ◽  
Compact Approximation ◽  
Weak Operator ◽  
Norm Attaining Operators ◽  
Norm Attaining ◽  
Necessary And Sufficient

Abstract In this article, we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in $\mathcal {L}(E, F)$ . By using a theorem due to Pfitzner on James boundaries, we show that if there exists a relatively compact set K of $\mathcal {L}(E, F)$ (in the weak operator topology) such that $0$ is an element of its closure (in the weak operator topology) but it is not in its norm-closed convex hull, then we can guarantee the existence of an operator that does not attain its norm. This allows us to provide the following generalisation of results due to Holub and Mujica. If E is a reflexive space, F is an arbitrary Banach space and the pair $(E, F)$ has the (pointwise-)bounded compact approximation property, then the following are equivalent: (i) $\mathcal {K}(E, F) = \mathcal {L}(E, F)$ ; (ii) Every operator from E into F attains its norm; (iii) $(\mathcal {L}(E,F), \tau _c)^* = (\mathcal {L}(E, F), \left \Vert \cdot \right \Vert )^*$ , where $\tau _c$ denotes the topology of compact convergence. We conclude the article by presenting a characterisation of the Schur property in terms of norm-attaining operators.

Groups with Finite Dimensional Irreducible Multiplier Representations

1985 ◽  
Vol 37 (4) ◽  
pp. 635-643 ◽  
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.

Best Polynomial Approximation with Linear Constraints

1992 ◽  
Vol 44 (6) ◽  
pp. 1289-1302
Author(s):  
K. Pan ◽  
E. B. Saff
Keyword(s):  
Asymptotic Behavior ◽  
Polynomial Approximation ◽  
Asymptotic Relation ◽  
Sufficient Conditions ◽  
Linear Constraints ◽  
Necessary And Sufficient Conditions ◽  
Compact Set ◽  
Approximation By Polynomials ◽  
The Matrix ◽  
Necessary And Sufficient

AbstractLet A be a (k + 1) × (k + 1) nonzero matrix. For polynomials p ∈ Pn, set and . Let E ⊂ C be a compact set that does not separate the plane and f be a function continuous on E and analytic in the interior of E. Set and . Our goal is to study approximation to f on E by polynomials from Bn(A). We obtain necessary and sufficient conditions on the matrix A for the convergence En(A,f) → 0 to take place. These results depend on whether zero lies inside, on the boundary or outside E and yield generalizations of theorems of Clunie, Hasson and Saff for approximation by polynomials that omit a power of z. Let be such that . We also study the asymptotic behavior of the zeros of and the asymptotic relation between En(f) and En(A,f).

scholarly journals The regularity of the space of germs of Fréchet valued holomorphic functions and the mixed Hartog's theorem

2006 ◽  
Vol 99 (1) ◽  
pp. 119 ◽  
Author(s):  
Thai Thuan Quang
Keyword(s):  
Holomorphic Function ◽  
Sufficient Conditions ◽  
Holomorphic Functions ◽  
Schwartz Space ◽  
Necessary And Sufficient Conditions ◽  
Fréchet Space ◽  
Compact Set ◽  
Frechet Space ◽  
Set Of Uniqueness ◽  
Necessary And Sufficient

It is shown that $H(K, F)$ is regular for every reflexive Fréchet space $F$ with the property ($\mathrm{LB}_\infty)$ where $K$ is a compact set of uniqueness in a Fréchet-Schwartz space $E$ such that $E \in (\Omega)$. Using this result we give necessary and sufficient conditions for a Fréchet space $F$, under which every separately holomorphic function on $K \times F^*$ is holomorphic, where $K$ is as above.

Left approximate identities in algebras of compact operators on Banach spaces

1986 ◽  
Vol 104 (1-2) ◽  
pp. 169-175 ◽  
Author(s):  
P. G. Dixon
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Finite Rank ◽  
Sufficient Conditions ◽  
Compact Operators ◽  
The Other ◽  
Compact Set ◽  
Approximate Identities ◽  
Necessary And Sufficient

SynopsisWe study the existence of left approximate units, left approximate identities and bounded left approximate identities in the algebras (X)of all compact operators on a Banach space X and ℱ(X)− of all operators uniformly approximable by finite rank operators. In the case of bounded left approximate identities, necessary and sufficient conditions on X are obtained. In the other cases, sufficient conditions are obtained, together with an example of non-existence using a space constructed by Szankowski. The possibility of the sufficient conditions being also necessary depends on the question of whether every compact set is contained in the closure of the image of the unit ball under an operator in (X)(or ℱ(X)−). Sufficient conditions on X are obtained for this to be true, but it is conjectured that the answer for general X is negative.

REMARKS ON SELF-AFFINE FRACTALS WITH POLYTOPE CONVEX HULLS

Fractals ◽  
2010 ◽  
Vol 18 (04) ◽  
pp. 483-498 ◽  
Author(s):  
IBRAHIM KIRAT ◽  
ILKER KOCYIGIT
Keyword(s):  
Convex Hull ◽  
Lebesgue Measure ◽  
Spectral Radius ◽  
Sufficient Conditions ◽  
Convex Hulls ◽  
Compact Set ◽  
Joint Spectral Radius ◽  
Digit Set ◽  
Necessary And Sufficient ◽  
Digit Expansions

Suppose that the set [Formula: see text] of real n × n matrices has joint spectral radius less than 1. Then for any digit set D = {d1, …, dq} ⊂ ℝn, there exists a unique non-empty compact set [Formula: see text] satisfying [Formula: see text], which is typically a fractal set. We use the infinite digit expansions of the points of F to give simple necessary and sufficient conditions for the convex hull of F to be a polytope. Additionally, we present a technique to determine the vertices of such polytopes. These answer some of the related questions of Strichartz and Wang, and also enable us to approximate the Lebesgue measure of such self-affine sets. To show the use of our results, we also give several examples including the Levy dragon and the Heighway dragon.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Budgeting in Socio-economic Development Strategies (Raising a Problem)

2008 ◽  
pp. 134-151
Author(s):  
A. Shastitko ◽  
M. Ovchinnikov
Keyword(s):  
Economic Policy ◽  
Economic Development ◽  
Social Change ◽  
Eastern Europe ◽  
Statistical Data ◽  
Sufficient Conditions ◽  
Transitional Period ◽  
Socio Economic Development ◽  
Necessary And Sufficient ◽  
Economic Development Strategies

The article proposes an approach to the analysis of social change and contributes to the clarification of concepts of economic policy. It deals in particular with the notion of "change of system". The author considers positive and normative aspects of the analysis of capitalist and socialist systems. The necessary and sufficient conditions for the system to be changed are introduced, their fulfillment is discussed drawing upon the historical and statistical data. The article describes both economic and political peculiarities of the transitional period in different countries, especially in Eastern Europe.

Inventory optimization considering the vehicle capacity and specifics of cash flows at rental storage sites

2020 ◽  
pp. 77-90
Author(s):  
V.D. Gerami ◽  
I.G. Shidlovskii
Keyword(s):  
Supply Chain ◽  
Inventory Management ◽  
Sufficient Conditions ◽  
Cash Flows ◽  
Management Systems ◽  
Cargo Capacity ◽  
Special Modification ◽  
The Cost ◽  
Necessary And Sufficient ◽  
Vehicle Capacity

The article presents a special modification of the EOQ formula and its application to the accounting of the cargo capacity factor for the relevant procedures for optimizing deliveries when renting storage facilities. The specified development will allow managers to take into account the following process specifics in the format of a simulated supply chain when managing inventory. First of all, it will allow considering the most important factor of cargo capacity when optimizing stocks. Moreover, this formula will make it possible to find the optimal strategy for the supply of goods if, also, it is necessary to take into account the combined effect of several factors necessary for practice, which will undoubtedly affect decision-making procedures. Here we are talking about the need for additional consideration of the following essential attributes of the simulated cash flow of the supply chain: 1) time value of money; 2) deferral of payment of the cost of the order; 3) pre-agreed allowable delays in the receipt of revenue from goods sold. Developed analysis and optimization procedures have been implemented to models of this type that are interesting and important for a business. This — inventory management systems, the format of which is related to the special concept of efficient supply. We are talking about models where the presence of the specified delays for the outgoing cash flows allows you to pay for the order and the corresponding costs of the supply chain from the corresponding revenue on the re-order interval. Accordingly, the necessary and sufficient conditions are established based on which managers will be able to identify models of the specified type. The purpose of the article is to draw the attention of managers to real opportunities to improve the efficiency of inventory management systems by taking into account these factors for a simulated supply chain.

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