On the numerical index with respect to an operator Kadets, Vladimir Martín Suárez, Miguel Meri De La Maza, Francisco Javier Quero de la Rosa, Alicia The research of the first author is done in frames of Ukrainian Ministry of Science and Education Research Pro gram 0118U002036, and it was partially done during his stay in the University of Granada which was supported by the project MTM2015-65020-P (MINECO/FEDER, UE). Research of second, third, and fifth authors is supported by projects MTM2015-65020-P (MINECO/FEDER, UE), PGC2018-093794-B-I00 (MCIU/AEI/FEDER, UE), and FQM-185 (Junta de Andalucía/FEDER, UE). The fourth author acknowledges financial support from the Spanish Ministry of Economy and Com petitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554). 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∈L(X,Y) (the space of all bounded linear operators from X to Y), the numerical index with respect to G, nG(X,Y), is the greatest constant k≥0 such that k∥T∥≤infδ>0sup{|y∗(Tx)|:y∗∈Y∗,x∈X,∥y∗∥=∥x∥=1,Rey∗(Gx)>1−δ} for every T∈L(X,Y). Equivalently, nG(X,Y) is the greatest constant k≥0 such that max∣∣w∣∣=1∥G+wT∥≥1+k∥T∥ for all T∈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 N(L(X,Y)) of the values of the numerical indices with respect to all norm-one operators in L(X,Y). For instance, N(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 N(L(X,ℓp))⊆[0,Mp]andN(L(ℓp,Y))⊆[0,Mp] for 1<p<∞ and for all real Banach spaces X and Y, where Mp=supt∈[0,1]∣∣tp−1−t∣∣1+tp. For complex Hilbert spaces H1, H2 of dimension greater than 1, N(L(H1,H2))⊆{0,1/2} and the value 1/2 is taken if and only if H1 and H2 are isometrically isomorphic. Moreover, N(L(X,H))⊆[0,1/2] and N(L(H,Y))⊆[0,1/2] when H is a complex infinite-dimensional Hilbert space and X and Y are arbitrary complex Banach spaces. Also, N(L(L1(μ1),L1(μ2)))⊆{0,1} and N(L(L∞(μ1),L∞(μ2)))⊆{0,1} for arbitrary σ-finite measures μ1 and μ2, 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 c0-, ℓ1-, or ℓ∞-sums of Banach spaces, composition operators on some vector-valued function spaces, taking the adjoint to an operator, and composition of operators. 2021-10-06T10:39:17Z 2021-10-06T10:39:17Z 2019-05-29 info:eu-repo/semantics/article Publisher version: Kadets, V., Martín, M., Merí, J., Pérez, A., & Quero, A. (2020). On the numerical index with respect to an operator. Dissertationes Mathematicae, 547, 1-58.[DOI: 10.4064/dm805-9-2019] http://hdl.handle.net/10481/70686 eng http://creativecommons.org/licenses/by-nc-nd/3.0/es/ info:eu-repo/semantics/openAccess Atribución-NoComercial-SinDerivadas 3.0 España