On the numerical index with respect to an operator
Volume 547 / 2020
Abstract
The aim of this paper is to study the numerical index with respect to an operator between Banach spaces. Given Banach spaces $X$ and $Y$, and a norm-one operator $G\in \mathcal{L}(X,Y)$ (the space of all bounded linear operators from $X$ to $Y$), the numerical index with respect to $G$, $n_G(X,Y)$, is the greatest constant $k\geq 0$ such that $$ k\|T\|\leq \inf_{\delta \gt 0} \sup\{|y^\ast(Tx)|\colon y^\ast\in Y^\ast,\,x\in X,\,\|y^\ast\|=\|x\|=1,\,\operatorname{Re} y^\ast(Gx) \gt 1-\delta\} $$ for every $T\in \mathcal{L}(X,Y)$. Equivalently, $n_G(X,Y)$ is the greatest constant $k\geq 0$ such that $$ \max_{|w|=1}\|G+wT\|\geq 1 + k \|T\| $$ for all $T\in \mathcal{L}(X,Y)$. Here, we first provide some tools to study the numerical index with respect to $G$. Next, 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 in $\mathcal{L}(X,Y)$. For instance, $\mathcal{N}(\mathcal{L}(X,Y))=\{0\}$ when $X$ or $Y$ is a real Hilbert space of dimension greater than 1 and also when $X$ or $Y$ is the space of bounded or compact operators on an infinite-dimensional real Hilbert space. In the real case $$ \mathcal{N}(\mathcal{L}(X,\ell_p))\subseteq [0,M_p] \quad \text{and} \quad \mathcal{N}(\mathcal{L}(\ell_p,Y))\subseteq [0,M_p] $$ for $1 \lt p \lt \infty$ and for all real Banach spaces $X$ and $Y$, where $M_p=\sup_{t\in[0,1]}\frac{|t^{p-1}-t|}{1+t^p}$. For complex Hilbert spaces $H_1$, $H_2$ of dimension greater than 1, $\mathcal{N}(\mathcal{L}(H_1,H_2))\subseteq \{0,1/2\}$ and the value $1/2$ is taken if and only if $H_1$ and $H_2$ are isometrically isomorphic. Moreover, $\mathcal{N}(\mathcal{L}(X,H))\subseteq [0,1/2]$ 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. Also, $\mathcal{N}(\mathcal{L}(L_1(\mu_1),L_1(\mu_2)))\subseteq \{0,1\}$ 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 from a Banach space to itself 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, such as constructing diagonal operators between $c_0$-, $\ell_1$-, or $\ell_\infty$-sums of Banach spaces, composition operators on some vector-valued function spaces, taking the adjoint to an operator, and composition of operators.
