Historical Introduction: A Walk on the Results for Banach Spaces with Numerical Index 1

2018 ◽  
pp. 1-36
Author(s):  
Vladimir Kadets ◽  
Miguel Martín ◽  
Javier Merí ◽  
Antonio Pérez
Keyword(s):  
Banach Spaces ◽  
Numerical Index

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.

scholarly journals Real Banach Spaces with Numerical Index 1

1999 ◽  
Vol 31 (2) ◽  
pp. 207-212 ◽  
Author(s):  
Ginés López ◽  
Miguel Martín ◽  
Rafael Payá
Keyword(s):  
Banach Spaces ◽  
Numerical Index

scholarly journals Finite-dimensional Banach spaces with numerical index zero

2004 ◽  
Vol 53 (5) ◽  
pp. 1279-1289 ◽  
Author(s):  
Miguel Martin ◽  
Javier Meri ◽  
Angel Rodriguez-Palacios
Keyword(s):  
Banach Spaces ◽  
Index Zero ◽  
Finite Dimensional ◽  
Numerical Index

scholarly journals Two-dimensional Banach spaces with polynomial numerical index zero

2009 ◽  
Vol 430 (8-9) ◽  
pp. 2488-2500 ◽  
Author(s):  
Domingo García ◽  
Bogdan C. Grecu ◽  
Manuel Maestre ◽  
Miguel Martín ◽  
Javier Merí
Keyword(s):  
Banach Spaces ◽  
Two Dimensional ◽  
Index Zero ◽  
Numerical Index

scholarly journals Convexity and smoothness of Banach spaces with numerical index one

2009 ◽  
Vol 53 (1) ◽  
pp. 163-182 ◽  
Author(s):  
Vladimir Kadets ◽  
Miguel Martín ◽  
Javier Merí ◽  
Rafael Payá
Keyword(s):  
Banach Spaces ◽  
Numerical Index

scholarly journals Numerical index of absolute sums of Banach spaces

2011 ◽  
Vol 375 (1) ◽  
pp. 207-222 ◽  
Author(s):  
Miguel Martín ◽  
Javier Merí ◽  
Mikhail Popov ◽  
Beata Randrianantoanina
Keyword(s):  
Banach Spaces ◽  
Numerical Index

scholarly journals On the numerical index of polyhedral Banach spaces

2019 ◽  
Vol 577 ◽  
pp. 121-133 ◽  
Author(s):  
Debmalya Sain ◽  
Kallol Paul ◽  
Pintu Bhunia ◽  
Santanu Bag
Keyword(s):  
Banach Spaces ◽  
Numerical Index
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