On the numerical index with respect to an operator

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
V. Kadets, Miguel Martín, Javier Merí, Antonio Pérez, Alicia Quero
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引用次数: 3

Abstract

Given Banach spaces $X$ and $Y$, and a norm-one operator $G\in \mathcal{L}(X,Y)$, the numerical index with respect to $G$, $n_G(X,Y)$, is the greatest constant $k\geq 0$ such that $$\max_{|w|=1}\|G+wT\|\geq 1 + k \|T\|$$ for all $T\in \mathcal{L}(X,Y)$. We present some results on the set $\mathcal{N}(\mathcal{L}(X,Y))$ of the values of the numerical indices with respect to all norm-one operators on $\mathcal{L}(X,Y)$. We show that $\mathcal{N}(\mathcal{L}(X,Y))=\{0\}$ when $X$ or $Y$ is a real Hilbert space of dimension greater than one and also when $X$ or $Y$ is the space of bounded or compact operators on an infinite-dimensional real Hilbert space. For complex Hilbert spaces $H_1$, $H_2$ of dimension greater than one, we show that $\mathcal{N}(\mathcal{L}(H_1,H_2))\subseteq \{0,1/2\}$ and the value $1/2$ is taken if and only if $H_1$ and $H_2$ are isometrically isomorphic. Besides, $\mathcal{N}(\mathcal{L}(X,H))\subseteq [0,1/2]$ and $\mathcal{N}(\mathcal{L}(H,Y))\subseteq [0,1/2]$ when $H$ is a complex infinite-dimensional Hilbert space and $X$ and $Y$ are arbitrary complex Banach spaces. We also show that $\mathcal{N}(\mathcal{L}(L_1(\mu_1),L_1(\mu_2)))\subseteq \{0,1\}$ and $\mathcal{N}(\mathcal{L}(L_\infty(\mu_1),L_\infty(\mu_2)))\subseteq \{0,1\}$ for arbitrary $\sigma$-finite measures $\mu_1$ and $\mu_2$, in both the real and the complex cases. Also, we show that the Lipschitz numerical range of Lipschitz maps can be viewed as the numerical range of convenient bounded linear operators with respect to a bounded linear operator. Further, we provide some results which show the behaviour of the value of the numerical index when we apply some Banach space operations, as constructing diagonal operators between $c_0$-, $\ell_1$-, or $\ell_\infty$-sums of Banach spaces, composition operators on some vector-valued function spaces, and taking the adjoint to an operator.
关于一个运算符的数值索引
给定Banach空间$X$和$Y$,以及范数一算子$G\in\mathcal{L}(X,Y)$,关于$G$,$n_G(X,Y)$的数值索引是最大常数$k\geq0$,使得$$$\max_{|w|=1}\|G+wT\|\geq1+k\|T\|$$对于所有$T\in\math cal{L}(X,Y)$。我们给出了关于$\mathcal{L}(X,Y)$上所有范数一算子的数值索引值的集合$\mathical{N}(\mathcal{L})(X,Y))$的一些结果。我们证明了当$X$或$Y$是维数大于1的实Hilbert空间时,以及当$X美元或$Y美元是无穷维实Hilbert空上有界或紧致算子的空间时,$\mathcal{N}(\mathcal{L}(X,Y))=\{0\}$。对于维数大于1的复Hilbert空间$H_1$,$H_2$,我们证明了$\mathcal{N}(\mathcal{L}(H_1,H_2))\substeq\{0,1/2\}$和值$1/2$是当且仅当$H_1$和$H_2$等距同构的。此外,当$H$是复无穷维Hilbert空间,$X$和$Y$是任意复Banach空间时,$\mathcal{N}(\mathcal{L}(X,H))\substeq[0,1/2]$和$\mathical{N}(\mathcal{L}(H,Y))\ssubsteq[0.1/2]$。我们还证明了在实数和复数情况下,任意$\sigma$-有限测度$\mu_1$和$\mu_2$的$\mathcal{N}(\mathcal{L}(L_1(\mu_1),L_1(\ mu_2))\substeq\{0,1\}$和$\mathcal{N}。此外,我们还证明了Lipschitz映射的Lipschitz-数值范围可以看作是方便的有界线性算子相对于有界线性运算符的数值范围。此外,我们还提供了一些结果,表明了当我们应用一些Banach空间操作时,数值指数的值的行为,如构造Banach空间的$c_0$-、$\ell_1$-或$\ell_\infty$-和之间的对角算子,一些向量值函数空间上的复合算子,以及取算子的伴随。
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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