# On the numerical index with respect to an operator

Research paper by **Vladimir Kadets, Miguel Martin, Javier Meri, Antonio Perez, Alicia Quero**

Indexed on: **29 Dec '19**Published on: **29 May '19**Published in: **arXiv - Mathematics - Functional Analysis**

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#### 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.