scholarly journals Restriction of Families of Conversion Operators in Lp Spaces

2021 ◽  
Vol 27 (3) ◽  
pp. 449-460
Author(s):  
A. D. Nakhman
Keyword(s):  
Fourier Coefficients ◽  
Continuous Functions ◽  
Parameter Family ◽  
Maximal Operators ◽  
Limiting Behavior ◽  
Lp Spaces ◽  
Almost Everywhere ◽  
Spaces Of Continuous Functions

We study a one-parameter family of convolutional operators acting in Lebesgue Lp spaces. The case of integral kernels given by the Fourier coefficients is considered. It is established that the condition of the coefficients being quasiconvex ensures the boundedness of the corresponding maximal operators. The limiting behavior of families in the metrics of spaces of continuous functions and Lp, p ≥ 1, classes is studied, and their convergence is obtained almost everywhere. The ways of possible generalizations and distributions are indicated.

scholarly journals Averaging operators in non commutative Lp spaces I

1983 ◽  
Vol 24 (1) ◽  
pp. 71-74 ◽  
Author(s):  
Christopher Barnett
Keyword(s):  
Von Neumann Algebra ◽  
Measure Theory ◽  
Continuous Functions ◽  
Unbounded Operators ◽  
Averaging Operators ◽  
Von Neumann ◽  
Lp Spaces ◽  
Finite Von Neumann Algebra ◽  
Spaces Of Continuous Functions ◽  
Finite Trace

The origin of the theory of averaging operators is explained in [1]. The theory has been developed on spaces of continuous functions that vanish at infinity by Kelley in [3] and on the Lp spaces of measure theory by Rota [5]. The motivation for this paper arose out of the latter paper. The aim of this paper is to prove a generalisation of Rota's main representation theorem (every average is a conditional expectation) in the context of a ‘non commutative integration’. This context is as follows. Let be a finite von Neumann algebra and ϕ a faithful normal finite trace on such that ϕ(I) = 1, where I is the identity of . We can construct the Banach spaces Lp (, ϕ), where 1 ≤ p < °, with norm ∥x∥p = ϕ(÷x÷p)1/p, of possibly unbounded operators affiliated with , as in [9]. We note that is dense in Lp(, ϕ). These spaces share many of the features of the Lp spaces of measure theory; indeed if is abelian then Lp(,ϕ) is isometrically isomorphic to Lp of some measure space.

scholarly journals SUBPROJECTIVITY OF PROJECTIVE TENSOR PRODUCTS OF BANACH SPACES OF CONTINUOUS FUNCTIONS

2021 ◽  
pp. 1-14
Author(s):  
R.M. CAUSEY
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Tensor Products ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Spaces Of Continuous Functions

Abstract Galego and Samuel showed that if K, L are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is c0-saturated if and only if it is subprojective if and only if K and L are both scattered. We remove the hypothesis of metrizability from their result and extend it from the case of the twofold projective tensor product to the general n-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces K1, …, K n , $\widehat{\otimes}_{\pi, i=1}^n C(K_i)$ is c0-saturated if and only if it is subprojective if and only if each K i is scattered.

scholarly journals BOUNDED LINEAR OPERATORS ON SPACES IN NORMED DUALITY

2007 ◽  
Vol 49 (1) ◽  
pp. 145-154
Author(s):  
BRUCE A. BARNES
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Integral Operators ◽  
Continuous Functions ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Closed Range ◽  
Bounded Linear ◽  
Spaces Of Continuous Functions

Abstract.LetTbe a bounded linear operator on a Banach spaceW, assumeWandYare in normed duality, and assume thatThas adjointT†relative toY. In this paper, conditions are given that imply that for all λ≠0, λ−Tand λ −T†maintain important standard operator relationships. For example, under the conditions given, λ −Thas closed range if, and only if, λ −T†has closed range.These general results are shown to apply to certain classes of integral operators acting on spaces of continuous functions.

scholarly journals Semi-smooth Points in Some Classical Function Spaces

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Beata Derȩgowska ◽  
Beata Gryszka ◽  
Karol Gryszka ◽  
Paweł Wójcik
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
Spaces Of Continuous Functions ◽  
Classical Function ◽  
Necessary And Sufficient ◽  
Vector Valued

AbstractThe investigations of the smooth points in the spaces of continuous function were started by Banach in 1932 considering function space $$\mathcal {C}(\Omega )$$ C ( Ω ) . Singer and Sundaresan extended the result of Banach to the space of vector valued continuous functions $$\mathcal {C}(\mathcal {T},E)$$ C ( T , E ) , where $$\mathcal {T}$$ T is a compact metric space. The aim of this paper is to present a description of semi-smooth points in spaces of continuous functions $$\mathcal {C}_0(\mathcal {T},E)$$ C 0 ( T , E ) (instead of smooth points). Moreover, we also find necessary and sufficient condition for semi-smoothness in the general case.

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