A note on Banach’s results concerning homogeneous polynomials associated with nonnegative tensors

2020 ◽  
Author(s):  
Chang Liang ◽  
Yuning Yang
Keyword(s):  
Homogeneous Polynomials ◽  
Nonnegative Tensors

An Inequality for Homogeneous Polynomials on ℝn

2005 ◽  
Vol 112 (3) ◽  
pp. 264-266
Author(s):  
Luo Xuebo ◽  
Zhu-Jun Zheng
Keyword(s):  
Homogeneous Polynomials

scholarly journals Some inequalities on the spectral radius of nonnegative tensors

2020 ◽  
Vol 18 (1) ◽  
pp. 262-269
Author(s):  
Chao Ma ◽  
Hao Liang ◽  
Qimiao Xie ◽  
Pengcheng Wang
Keyword(s):  
Spectral Radius ◽  
Nonnegative Tensors

Abstract The eigenvalues and the spectral radius of nonnegative tensors have been extensively studied in recent years. In this paper, we investigate the analytic properties of nonnegative tensors and give some inequalities on the spectral radius.

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.

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