scholarly journals On a second numerical index for Banach spaces

2019 ◽  
Vol 150 (2) ◽  
pp. 1003-1051 ◽  
Author(s):  
Sun Kwang Kim ◽  
Han Ju Lee ◽  
Miguel Martín ◽  
Javier Merí
Keyword(s):  
Banach Spaces ◽  
Lie Algebra ◽  
Hilbert Spaces ◽  
Unconditional Basis ◽  
Numerical Radius ◽  
Best Constant ◽  
The One ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Numerical Index

AbstractWe introduce a second numerical index for real Banach spaces with non-trivial Lie algebra, as the best constant of equivalence between the numerical radius and the quotient of the operator norm modulo the Lie algebra. We present a number of examples and results concerning absolute sums, duality, vector-valued function spaces…which show that, in many cases, the behaviour of this second numerical index differs from the one of the classical numerical index. As main results, we prove that Hilbert spaces have second numerical index one and that they are the only spaces with this property among the class of Banach spaces with one-unconditional basis and non-trivial Lie algebra. Besides, an application to the Bishop-Phelps-Bollobás property for the numerical radius is given.

scholarly journals THE POLYNOMIAL NUMERICAL INDEX OF A BANACH SPACE

2006 ◽  
Vol 49 (1) ◽  
pp. 39-52 ◽  
Author(s):  
Yun Sung Choi ◽  
Domingo Garcia ◽  
Sung Guen Kim ◽  
Manuel Maestre
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Banach Spaces ◽  
Compact Space ◽  
Linear Operators ◽  
Homogeneous Polynomials ◽  
Numerical Radius ◽  
Disc Algebra ◽  
Multilinear Maps ◽  
Numerical Index

AbstractIn this paper, we introduce the polynomial numerical index of order $k$ of a Banach space, generalizing to $k$-homogeneous polynomials the ‘classical’ numerical index defined by Lumer in the 1970s for linear operators. We also prove some results. Let $k$ be a positive integer. We then have the following:(i) $n^{(k)}(C(K))=1$ for every scattered compact space $K$.(ii) The inequality $n^{(k)}(E)\geq k^{k/(1-k)}$ for every complex Banach space $E$ and the constant $k^{k/(1-k)}$ is sharp.(iii) The inequalities$$ n^{(k)}(E)\leq n^{(k-1)}(E)\leq\frac{k^{(k+(1/(k-1)))}}{(k-1)^{k-1}}n^{(k)}(E) $$for every Banach space $E$.(iv) The relation between the polynomial numerical index of $c_0$, $l_1$, $l_{\infty}$ sums of Banach spaces and the infimum of the polynomial numerical indices of them.(v) The relation between the polynomial numerical index of the space $C(K,E)$ and the polynomial numerical index of $E$.(vi) The inequality $n^{(k)}(E^{**})\leq n^{(k)}(E)$ for every Banach space $E$.Finally, some results about the numerical radius of multilinear maps and homogeneous polynomials on $C(K)$ and the disc algebra are given.

Endpoint Regularity of Multisublinear Fractional Maximal Functions

2017 ◽  
Vol 60 (3) ◽  
pp. 586-603 ◽  
Author(s):  
Feng Liu ◽  
Huoxiong Wu
Keyword(s):  
Maximal Operator ◽  
Maximal Functions ◽  
Maximal Operators ◽  
One Dimensional ◽  
Regularity Properties ◽  
The One ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Littlewood Maximal Operator

AbstractIn this paper we investigate the endpoint regularity properties of the multisublinear fractional maximal operators, which include the multisublinear Hardy–Littlewood maximal operator. We obtain some new bounds for the derivative of the one-dimensional multisublinear fractional maximal operators acting on the vector-valued function with all ƒ j being BV-functions.

CONTRACTIONS OF LIE ALGEBRA REPRESENTATIONS

1999 ◽  
Vol 11 (10) ◽  
pp. 1179-1207 ◽  
Author(s):  
U. CATTANEO ◽  
W. F. WRESZINSKI
Keyword(s):  
General Theory ◽  
Banach Spaces ◽  
Lie Algebra ◽  
Hilbert Spaces ◽  
3 Dimensional ◽  
Algebra Representation ◽  
Carrier Space ◽  
Definition Of ◽  
Lie Algebra Representations ◽  
Main Distinguishing Feature

A theory of contractions of Lie algebra representations on complex Hilbert spaces is proposed, based on Trotter's theory of approximating sequences of Banach spaces. Its main distinguishing feature is a careful definition of the carrier space of the limit Lie algebra representation. A set of quite general conditions on the contracting representations, satisfied in all known examples, is proven to be sufficient for the existence of such a representation. In order to show how natural the suggested framework is, the general theory is applied to the contraction of [Formula: see text] into the Lie algebra [Formula: see text] of the 3-dimensional Heisenberg group and to the related study of the limit N→∞ of a quantum system of N identical two-level particles.

REFINEMENTS OF THE CONTINUOUS TRIANGLE INEQUALITY FOR THE BOCHNER INTEGRAL IN HILBERT SPACES

2008 ◽  
Vol 01 (04) ◽  
pp. 521-533
Author(s):  
S. S. Dragomir
Keyword(s):  
Hilbert Spaces ◽  
Triangle Inequality ◽  
Bochner Integral ◽  
Numerical Radius ◽  
Operator Inequalities ◽  
Vector Valued Functions ◽  
Complex Valued ◽  
Vector Valued

Some refinements of the continuous triangle inequality for the Bochner integral of vector-valued functions in Hilbert spaces are given. Applications for norm and numerical radius operator inequalities are provided. A particular case of interest for complex-valued functions is pointed out as well.

scholarly journals On tensor products of nuclear operators in Banach spaces

2021 ◽  
Vol 14 (3) ◽  
pp. 187-205
Author(s):  
Oleg Reinov
Keyword(s):  
Hilbert Space ◽  
Abelian Group ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Compact Abelian Group ◽  
Convolution Operator ◽  
Fourier Coefficients ◽  
Schatten Classes ◽  
Nuclear Operators ◽  
Vector Valued

The following result of G. Pisier contributed to the appearance of this paper: if a convolution operator ★f : M(G) → C(G), where $G$ is a compact Abelian group, can be factored through a Hilbert space, then f has the absolutely summable set of Fourier coefficients. We give some generalizations of the Pisier's result to the cases of factorizations of operators through the operators from the Lorentz-Schatten classes Sp,q in Hilbert spaces both in scalar and in vector-valued cases. Some applications are given.

scholarly journals POLYNOMIAL NUMERICAL INDEX FOR SOME COMPLEX VECTOR-VALUED FUNCTION SPACES

2007 ◽  
Vol 59 (4) ◽  
pp. 455-474 ◽  
Author(s):  
Y. S. Choi ◽  
D. Garcia ◽  
M. Maestre ◽  
M. Martin
Keyword(s):  
Function Spaces ◽  
Complex Vector ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Numerical Index

On the Basis Problem for Vector Valued Function Spaces

1956 ◽  
Vol 8 ◽  
pp. 417-422 ◽  
Author(s):  
H. W. Ellis
Keyword(s):  
Banach Spaces ◽  
Function Spaces ◽  
Length Function ◽  
General Measure ◽  
Basis Problem ◽  
Measure Spaces ◽  
Vector Valued Function ◽  
Vector Valued

1. Introduction. In a recent paper (2) Halperin and the author considered separable Banach spaces Lλ of real valued functions on general measure spaces and proved the existence of 1-regular (§2) Haar or σ-Haar bases when λ was the classical p-norm or any levelling length function (3) and, more generally, of K-regular Haar or σ-Haar bases when λ was a continuous length function satisfying certain additional conditions (2, Theorem 3.2).

On l1 subspaces of Orlicz vector-valued function spaces

1987 ◽  
Vol 101 (1) ◽  
pp. 107-112 ◽  
Author(s):  
Fernando Bombal
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Function Spaces ◽  
Complete Characterization ◽  
Complemented Subspace ◽  
Vector Valued Function ◽  
Vector Valued ◽  
General Banach Space

The purpose of this paper is to characterize the Orlicz vector-valued function spaces containing a copy or a complemented copy of l1. Pisier proved in [13] that if a Banach space E contains no copy of l1, then the space Lp(S, Σ, μ, E) does not contain it either, for 1 < p < ∞. We extend this result to the case of Orlicz vector valued function spaces, by reducing the problem to the situation considered by Pisier. Next, we pass to study the problem of embedding l1 as a complemented subspace of LΦ(E). We obtain a complete characterization when E is a Banach lattice and only partial results in case of a general Banach space. We use here in a crucial way a result of E. Saab and P. Saab concerning the embedding of l1 as a complemented subspace of C(K, E), the Banach space of all the E-valued continuous functions on the compact Hausdorff space K (see [14]). Finally, we use these results to characterize several classes of Banach spaces for which LΦ(E) has some Banach space properties, namely the reciprocal Dunford-Pettis property and Pelczyński's V property.

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