On the Maximal Operator Ideal Associated with a Tensor Norm Defined by Interpolation Spaces

2010 ◽  
Vol 53 (4) ◽  
pp. 690-705
Author(s):  
M. E. Puerta ◽  
G. Loaiza
Keyword(s):  
Maximal Operator ◽  
Interpolation Space ◽  
Sufficient Conditions ◽  
Classical Approach ◽  
Operator Ideals ◽  
Metric Properties ◽  
Real Method ◽  
Tensor Norm ◽  
Tensor Norms ◽  
Necessary And Sufficient

AbstractThe classical approach to studying operator ideals using tensor norms mainly focuses on those tensor norms and operator ideals defined by means of ℓp spaces. In a previous paper, an interpolation space, defined via the real method and using ℓp spaces, was used to define a tensor norm, and the associated minimal operator ideals were characterized. In this paper, the next natural step is taken, that is, the corresponding maximal operator ideals are characterized. As an application, necessary and sufficient conditions for the coincidence of the maximal and minimal ideals are given. Finally, the previous results are used in order to find some new metric properties of the mentioned tensor norm.

scholarly journals Operator ideals and tensor norms defined by a sequence space

2004 ◽  
Vol 69 (3) ◽  
pp. 499-517 ◽  
Author(s):  
J.A. López Molina ◽  
M.J. Rivera
Keyword(s):  
Classical Theory ◽  
Sequence Space ◽  
Maximal Operator ◽  
Dimensional Structure ◽  
Operator Ideals ◽  
Finite Dimensional ◽  
Tensor Norm ◽  
Tensor Norms

We study the tensor norm defined by a sequence space λ and its minimal and maximal operator ideals associated in the sense of Defant and Floret. Our results extend the classical theory related to the tensor norms of Saphar [16]. They show the key role played by the finite dimensional structure of the ultrapowers of λ in this kind of problems.

scholarly journals Approximation properties of tensor norms and operator ideals for Banach spaces

2020 ◽  
Vol 18 (1) ◽  
pp. 1698-1708
Author(s):  
Ju Myung Kim
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Approximation Property ◽  
Approximation Properties ◽  
Operator Ideals ◽  
Finitely Generated ◽  
Tensor Norm ◽  
Tensor Norms

Abstract For a finitely generated tensor norm α \alpha , we investigate the α \alpha -approximation property ( α \alpha -AP) and the bounded α \alpha -approximation property (bounded α \alpha -AP) in terms of some approximation properties of operator ideals. We prove that a Banach space X has the λ \lambda -bounded α p , q {\alpha }_{p,q} -AP ( 1 ≤ p , q ≤ ∞ , 1 / p + 1 / q ≥ 1 ) (1\le p,q\le \infty ,1/p+1/q\ge 1) if it has the λ \lambda -bounded g p {g}_{p} -AP. As a consequence, it follows that if a Banach space X has the λ \lambda -bounded g p {g}_{p} -AP, then X has the λ \lambda -bounded w p {w}_{p} -AP.

Riesz potential in the local Morrey–Lorentz spaces and some applications

2020 ◽  
Vol 27 (4) ◽  
pp. 557-567
Author(s):  
Vagif S. Guliyev ◽  
Abdulhamit Kucukaslan ◽  
Canay Aykol ◽  
Ayhan Serbetci
Keyword(s):  
Maximal Operator ◽  
Riesz Potential ◽  
Sufficient Conditions ◽  
Lorentz Spaces ◽  
Necessary And Sufficient Conditions ◽  
Analytic Semigroups ◽  
Fractional Powers ◽  
Marcinkiewicz Operator ◽  
Fractional Maximal Operator ◽  
Necessary And Sufficient

AbstractIn this paper, the necessary and sufficient conditions are found for the boundedness of the Riesz potential {I_{\alpha}} in the local Morrey–Lorentz spaces {M_{p,q;{\lambda}}^{\mathrm{loc}}({\mathbb{R}^{n}})}. This result is applied to the boundedness of particular operators such as the fractional maximal operator, fractional Marcinkiewicz operator and fractional powers of some analytic semigroups on the local Morrey–Lorentz spaces {M_{p,q;{\lambda}}^{\mathrm{loc}}({\mathbb{R}^{n}})}.

A Modified Gradient Procedure for Adaptation in Nonlinear Control Systems

1970 ◽  
Vol 92 (2) ◽  
pp. 322-327 ◽  
Author(s):  
A. E. Pearson
Keyword(s):  
Nonlinear Control ◽  
Control Systems ◽  
Sufficient Conditions ◽  
Operating Conditions ◽  
Metric Properties ◽  
Quadratic Functionals ◽  
Squared Error ◽  
Time Changes ◽  
Error Index ◽  
Necessary And Sufficient

A modified gradient procedure is proposed for making discrete-time changes in the adjustable parameters of a continuous-time nonlinear control system during normal operating conditions. The algorithm employs the best available estimate of the unknown plant parameters as well as the estimates of disturbance, state, and output variables. The importance of the metric properties of a performance index is discussed, and the necessary and sufficient conditions for the integral squared error index to possess metric properties are derived. Theoretical conditions for the error correctiveness of the algorithm are formulated in terms of the constrained extrema of quadratic functionals.

Two-weight weak and extra-weak type inequalities for the one-sided maximal operator

1993 ◽  
Vol 123 (6) ◽  
pp. 1109-1118
Author(s):  
Pedro Ortega Salvador ◽  
Luboš Pick
Keyword(s):  
Maximal Operator ◽  
Weak Type ◽  
Orlicz Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weight Functions ◽  
The One ◽  
Image Position ◽  
Necessary And Sufficient

SynopsisLet be the one-sided maximal operator and let Ф be a convex non-decreasing function on (0, ∞), Ф(0) = 0. We present necessary and sufficient conditions on a couple of weight functions (σ, ϱ) such that the integral inqualities of weak typeand of extra-weak typehold. Our proofs do not refer to the theory of Orlicz spaces.

scholarly journals Necessary and Sufficient Conditions for the Boundedness of Dunkl-Type Fractional Maximal Operator in the Dunkl-Type Morrey Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-10
Author(s):  
Emin Guliyev ◽  
Ahmet Eroglu ◽  
Yagub Mammadov
Keyword(s):  
Maximal Operator ◽  
Morrey Space ◽  
Sufficient Conditions ◽  
Morrey Spaces ◽  
Necessary And Sufficient Conditions ◽  
Dunkl Operator ◽  
Generalized Shift Operator ◽  
Fractional Maximal Operator ◽  
Generalized Shift ◽  
Necessary And Sufficient

We consider the generalized shift operator, associated with the Dunkl operator , . We study the boundedness of the Dunkl-type fractional maximal operator in the Dunkl-type Morrey space , . We obtain necessary and sufficient conditions on the parameters for the boundedness , from the spaces to the spaces , , and from the spaces to the weak spaces , . As an application of this result, we get the boundedness of from the Dunkl-type Besov-Morrey spaces to the spaces , , , , , and .

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