scholarly journals THE POSITION OF IN

2013 ◽  
Vol 56 (2) ◽  
pp. 409-417 ◽  
Author(s):  
DANIELE PUGLISI
Keyword(s):  
Banach Spaces ◽  
Borel Structure ◽  
The Family

AbstractIn this paper we investigate the nature of family of pairs of separable Banach spaces (X, Y) such that $\mathcal{K}(X,Y)$ is complemented in $\mathcal{L}(X,Y)$. It is proved that the family of pairs (X,Y) of separable Banach spaces such that $\mathcal{K}(X,Y)$ is complemented in $\mathcal{L}(X,Y)$ is not Borel, endowed with the Effros–Borel structure.

scholarly journals The algebraic size of the family of injective operators

2017 ◽  
Vol 15 (1) ◽  
pp. 13-20 ◽  
Author(s):  
Luis Bernal-González
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Algebra ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
The Family ◽  
One To One ◽  
Algebra Of Operators

Abstract In this paper, a criterion for the existence of large linear algebras consisting, except for zero, of one-to-one operators on an infinite dimensional Banach space is provided. As a consequence, it is shown that every separable infinite dimensional Banach space supports a commutative infinitely generated free linear algebra of operators all of whose nonzero members are one-to-one. In certain cases, the assertion holds for nonseparable Banach spaces.

scholarly journals On the problem of non-smoothness of non-reflexive second conjugate spaces

1975 ◽  
Vol 12 (3) ◽  
pp. 407-416 ◽  
Author(s):  
Ivan Singer
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Subspace ◽  
Canonical Embedding ◽  
Reflexive Banach Spaces ◽  
Closed Linear Subspace ◽  
Conjugate Space ◽  
The Family

We prove that if E is a Banach space which has a subspace G such that the conjugate space G* contains a proper norm closed linear subspace V of characteristic 1, then E** is not smooth and there exist in πE(E) points of non-smoothness for E**, where πE: E → E** is the canonical embedding. We show that the spaces E having such a subspace G constitute a large proper subfamily of the family of all non-reflexive Banach spaces.

Atomic systems for operators

2019 ◽  
Vol 17 (01) ◽  
pp. 1850066 ◽  
Author(s):  
Khole Timothy Poumai ◽  
Shah Jahan
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Atomic System ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bessel Sequence ◽  
Atomic Systems ◽  
The Family ◽  
Necessary And Sufficient

Gavruta [L. Gavruta, Frames for operators, Appl. Comput. Harmon. Anal. 32 (2012) 139–144] introduced the notion of [Formula: see text]-frame and atomic system for an operator [Formula: see text] in Hilbert spaces. We extend these notions to Banach spaces and obtain various new results. A necessary and sufficient condition for the existence of an atomic system for an operator [Formula: see text] in a Banach space is given. Also, a characterization for the family of local atoms of subspaces of Banach spaces has been given. Further, we give methods to construct an atomic system for an operator [Formula: see text] from a given Bessel sequence and an [Formula: see text]-Bessel sequence. Finally, a result related to stability of atomic system for an operator [Formula: see text] in a Banach space has been given.

EQUIVALENCE OF SEMI-NORMS RELATED TO SUPER WEAKLY COMPACT OPERATORS

2021 ◽  
pp. 1-13
Author(s):  
KUN TU
Keyword(s):  
Banach Spaces ◽  
Quantitative Method ◽  
Approximation Property ◽  
Essential Norm ◽  
Compact Operators ◽  
Sequence Spaces ◽  
Weakly Compact ◽  
The Family ◽  
Compact Approximation ◽  
Weakly Compact Operators

Abstract We study super weakly compact operators through a quantitative method. We introduce a semi-norm $\sigma (T)$ of an operator $T:X\to Y$ , where X, Y are Banach spaces, the so-called measure of super weak noncompactness, which measures how far T is from the family of super weakly compact operators. We study the equivalence of the measure $\sigma (T)$ and the super weak essential norm of T. We prove that Y has the super weakly compact approximation property if and and only if these two semi-norms are equivalent. As an application, we construct an example to show that the measures of T and its dual $T^*$ are not always equivalent. In addition we give some sequence spaces as examples of Banach spaces having the super weakly compact approximation property.

scholarly journals REPRESENTATIONS OF IDEALS IN POLISH GROUPS AND IN BANACH SPACES

2015 ◽  
Vol 80 (4) ◽  
pp. 1268-1289 ◽  
Author(s):  
PIOTR BORODULIN–NADZIEJA ◽  
BARNABÁS FARKAS ◽  
GRZEGORZ PLEBANEK
Keyword(s):  
Banach Space ◽  
Abelian Group ◽  
Banach Spaces ◽  
Open Problems ◽  
Polish Groups ◽  
Polish Group ◽  
Focus On The Family ◽  
The Family ◽  
Null Ideal ◽  
Defining Sequence

AbstractWe investigate ideals of the form {A⊆ω: Σn∈Axnis unconditionally convergent} where (xn)n∈ωis a sequence in a Polish group or in a Banach space. If an ideal onωcan be seen in this form for some sequence inX, then we say that it is representable inX.After numerous examples we show the following theorems: (1) An ideal is representable in a Polish Abelian group iff it is an analytic P-ideal. (2) An ideal is representable in a Banach space iff it is a nonpathological analytic P-ideal.We focus on the family of ideals representable inc0. We characterize this property via the defining sequence of measures. We prove that the trace of the null ideal, Farah’s ideal, and Tsirelson ideals are not representable inc0, and that a tallFσP-ideal is representable inc0iff it is a summable ideal. Also, we provide an example of a peculiar ideal which is representable inℓ1but not in ℝ.Finally, we summarize some open problems of this topic.

GENERALIZATION OF THE FRAME OPERATOR AND THE CANONICAL DUAL FRAME TO BANACH SPACES

2008 ◽  
Vol 01 (04) ◽  
pp. 631-643 ◽  
Author(s):  
Diana T. Stoeva
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Dual Frame ◽  
Frame Operator ◽  
Hilbert Frames ◽  
The Family

Xd-frames for Banach spaces are generalization of Hilbert frames. In this paper we extend the concepts of frame operator and canonical dual to the case of Xd-frames. For a given Xd-frame {gi} for the Banach space X we define an Xd-frame map𝕊 : X → X* and determine conditions, which imply that 𝕊 is invertible and the family {𝕊-1gi} is an [Formula: see text]-frame for X* such that f = ∑gi(f)𝕊-1gi for every f ∈ X and g = ∑g(𝕊-1gi)gi for every g ∈ X*. If X is a Hilbert space and {gi} is a frame for X, then the ℓ2-frame map 𝕊 gives the frame operator S and the family {𝕊-1gi} coincides with the canonical dual of {gi}.

scholarly journals Variation and oscillation for harmonic operators in the inverse Gaussian setting

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Víctor Almeida ◽  
Jorge J. Betancor
Keyword(s):  
Banach Spaces ◽  
Fractional Derivatives ◽  
Riesz Transforms ◽  
Semigroups Of Operators ◽  
Inverse Gaussian ◽  
The Family ◽  
Derivatives Of ◽  
Martingale Cotype

<p style='text-indent:20px;'>We prove variation and oscillation <inline-formula><tex-math id="M1">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-inequalities associated with fractional derivatives of certain semigroups of operators and with the family of truncations of Riesz transforms in the inverse Gaussian setting. We also study these variational <inline-formula><tex-math id="M2">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-inequalities in a Banach-valued context by considering Banach spaces with the UMD-property and whose martingale cotype is fewer than the variational exponent. We establish <inline-formula><tex-math id="M3">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-boundedness properties for weighted difference involving the semigroups under consideration.</p>

scholarly journals Operator self-similar processes on Banach spaces

2006 ◽  
Vol 2006 ◽  
pp. 1-18 ◽  
Author(s):  
Mihaela T. Matache ◽  
Valentin Matache
Keyword(s):  
Hilbert Space ◽  
Stochastic Processes ◽  
Banach Spaces ◽  
Linear Space ◽  
Power Function ◽  
Linear Subspace ◽  
Multiplicative Group ◽  
The Family ◽  
Self Similar ◽  
Dense Linear Subspace

Operator self-similar (OSS) stochastic processes on arbitrary Banach spaces are considered. If the family of expectations of such a process is a spanning subset of the space, it is proved that the scaling family of operators of the process under consideration is a uniquely determined multiplicative group of operators. If the expectation-function of the process is continuous, it is proved that the expectations of the process have power-growth with exponent greater than or equal to 0, that is, their norm is less than a nonnegative constant times such a power-function, provided that the linear space spanned by the expectations has category 2 (in the sense of Baire) in its closure. It is shown that OSS processes whose expectation-function is differentiable on an interval (s0,∞), for some s0≥1, have a unique scaling family of operators of the form {sH:s>0}, if the expectations of the process span a dense linear subspace of category 2. The existence of a scaling family of the form {sH:s>0} is proved for proper Hilbert space OSS processes with an Abelian scaling family of positive operators.

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