Norm Convergence of Tn

1977 ◽  
Vol 29 (6) ◽  
pp. 1340-1344 ◽  
Author(s):  
Glenn R. Luecke
Keyword(s):  
Banach Space ◽  
Spectral Radius ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Necessary And Sufficient Conditions ◽  
Norm Convergence ◽  
Continuous Linear ◽  
Linear Transformations ◽  
Necessary And Sufficient

Throughout this paper X will denote a complex Banach space and all operators T will be assumed to be continuous linear transformations from X into X. If T is an operator then ┘(T), γ(T), and R(T) will denote the spectrum of T, the spectral radius of T, and range of T, respectively. This paper contains necessary and sufficient conditions for the (norm) convergence of {Tn} when T is an operator on X.

On the domain of Riesz mean in the space Ls

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 925-940 ◽  
Author(s):  
Medine Yeşilkayagil ◽  
Feyzi Başar
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Double Sequence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Double Sequences ◽  
Riesz Mean ◽  
Double Sequence Space ◽  
Necessary And Sufficient ◽  
Barrelled Space

Let 0 < s < ?. In this study, we introduce the double sequence space Rqt(Ls) as the domain of four dimensional Riesz mean Rqt in the space Ls of absolutely s-summable double sequences. Furthermore, we show that Rqt(Ls) is a Banach space and a barrelled space for 1 ? s < 1 and is not a barrelled space for 0 < s < 1. We determine the ?- and ?(?)-duals of the space Ls for 0 < s ? 1 and ?(bp)-dual of the space Rqt(Ls) for 1 < s < 1, where ? ? {p, bp, r}. Finally, we characterize the classes (Ls:Mu), (Ls:Cbp), (Rqt(Ls) : Mu) and (Rqt(Ls):Cbp) of four dimensional matrices in the cases both 0 < s < 1 and 1 ? s < 1 together with corollaries some of them give the necessary and sufficient conditions on a four dimensional matrix in order to transform a Riesz double sequence space into another Riesz double sequence space.

scholarly journals Complex extreme measurable selections

1995 ◽  
Vol 58 (2) ◽  
pp. 222-231
Author(s):  
Douglas Mupasiri
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Extreme Point ◽  
Measure Space ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Closed Unit Ball ◽  
Measurable Selections ◽  
Necessary And Sufficient

AbstractWe give a characterization of complex extreme measurable selections for a suitable set-valued map. We use this result to obtain necessary and sufficient conditions for a function to be a complex extreme point of the closed unit ball of Lp (ω, Σ, ν X), where (ω, σ, ν) is any positive, complete measure space, X is a separable complex Banach space, and 0 < p < ∞.

scholarly journals Modified cauchy kernels and functional calculus for operators on Banach space

1997 ◽  
Vol 63 (1) ◽  
pp. 91-99 ◽  
Author(s):  
Edwin Franks
Keyword(s):  
Banach Space ◽  
Functional Calculus ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Cauchy Kernel ◽  
Necessary And Sufficient ◽  
Banach Space Operators

AbstractIn Banach space operators with a bounded H∞ functional calculus, Cowling et al. provide some necessary and sufficient conditions for a type-ω operator to have a bounded H∞ functional calculus. We provide an alternate development of some of their ideas using a modified Cauchy kernel which is L1 with respect to the measure ]dz]/]z]. The method is direct and has the advantage that no transforms of the functions are necessary.

On the invertibility of length two elementary operators

2015 ◽  
Vol 30 ◽  
pp. 916-913
Author(s):  
Janko Bracic ◽  
Nadia Boudi
Keyword(s):  
Banach Space ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Elementary Operator ◽  
Bounded Linear Operators ◽  
Existence Of A Solution ◽  
Bounded Linear ◽  
Elementary Operators ◽  
Necessary And Sufficient

Let X be a complex Banach space and L(X) be the algebra of all bounded linear operators on X. For a given elementary operator P of length 2 on L(X), we determine necessary and sufficient conditions for the existence of a solution of the equation YP=0 in the algebra of all elementary operators on L(X). Our approach allows us to characterize some invertible elementary operators of length 2 whose inverses are elementary operators.

scholarly journals Balls intersection properties of Banach spaces

1992 ◽  
Vol 45 (2) ◽  
pp. 333-342 ◽  
Author(s):  
Dongjian Chen ◽  
Zhibao Hu ◽  
Bor-Luh Lin
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sufficient Conditions ◽  
Intersection Property ◽  
Necessary And Sufficient Conditions ◽  
Asplund Space ◽  
Necessary And Sufficient

Necessary and sufficient conditions for a Banach space with the Mazur intersection property to be an Asplund space are given. It is proved that Mazur intersection property is determined by the separable subspaces of the space. Corresponding problems for a space to have the ball-generated property are considered. Some comments on possible renorming so that a space having the Mazur intersection property are given.

scholarly journals Property (Bb) and Tensor product

Filomat ◽  
2013 ◽  
Vol 27 (7) ◽  
pp. 1297-1303
Author(s):  
M.H.M. Rashid ◽  
T. Prasad
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Space Operator ◽  
Weyl Spectrum ◽  
Riesz Operators ◽  
Necessary And Sufficient ◽  
Banach Space Operators ◽  
Satisfy Property

In this paper, we find necessary and sufficient conditions for Banach Space operator to satisfy the property (Bb). Then we obtain, if Banach Space operators A ? B(X)and B ? B(Y) satisfy property (Bb) implies A x B satisfies property (Bb) if and only if the B-Weyl spectrum identity ?BW(A x B) = ?BW(A)?(B) U ?BW(B)?(A) holds. Perturbations by Riesz operators are considered.

scholarly journals Measures of noncompactness and superposition operator in the space of regulated functions on an unbounded interval

2020 ◽  
Vol 114 (4) ◽  
Author(s):  
Szymon Dudek ◽  
Leszek Olszowy
Keyword(s):  
Banach Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Superposition Operator ◽  
Measures Of Noncompactness ◽  
Relative Compactness ◽  
Unbounded Interval ◽  
Bounded Functions ◽  
Regulated Functions ◽  
Necessary And Sufficient

Abstract In this paper, we formulate necessary and sufficient conditions for relative compactness in the space $$BG({\mathbb {R}}_+,E)$$ B G ( R + , E ) of regulated and bounded functions defined on $${\mathbb R}_+$$ R + with values in the Banach space E. Moreover, we construct four new measures of noncompactness in the space $$BG({\mathbb {R}}_+,E)$$ B G ( R + , E ) . We investigate their properties and we describe relations between these measures. We provide necessary and sufficient conditions so that the superposition operator (Niemytskii) maps $$BG({\mathbb {R}}_+,E)$$ B G ( R + , E ) into $$BG({\mathbb {R}}_+,E)$$ B G ( R + , E ) and, additionally, be compact.

scholarly journals Finding linking sets

2006 ◽  
Vol 2006 ◽  
pp. 1-13
Author(s):  
Martin Schechter ◽  
Kyril Tintarev
Keyword(s):  
Banach Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
General Definition ◽  
Definition Of ◽  
Necessary And Sufficient

We present the most general definition of the linking of sets in a Banach space and discuss necessary and sufficient conditions for sets to link.

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