Artykuły w formacie PDF dostępne są dla subskrybentów, którzy zapłacili za dostęp online, po podpisaniu licencji Licencja użytkownika instytucjonalnego. Czasopisma do 2009 są ogólnodostępne (bezpłatnie).

On the numerical index with respect to an operator

Tom 547 / 2020

Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez, Alicia Quero Dissertationes Mathematicae 547 (2020), 1-58 MSC: Primary 46B04; Secondary 46B20, 46B25, 46L05, 47A12, 47A30. DOI: 10.4064/dm805-9-2019 Opublikowany online: 2 January 2020

Streszczenie

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.

Autorzy

  • Vladimir KadetsSchool of Mathematics and Computer Sciences
    V. N. Karazin Kharkiv National University
    pl. Svobody 4
    61022 Kharkiv, Ukraine
    e-mail
  • Miguel MartínUniversidad de Granada
    Facultad de Ciencias
    Departamento de Análisis Matemático
    E-18071 Granada, Spain
    e-mail
  • Javier MeríUniversidad de Granada
    Facultad de Ciencias
    Departamento de Análisis Matemático
    E-18071 Granada, Spain
    e-mail
  • Antonio PérezInstituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)
    Campus Cantoblanco
    C/ Nicolás Cabrera, 13–15
    E-28049 Madrid, Spain
    e-mail
  • Alicia QueroUniversidad de Granada
    Facultad de Ciencias
    Departamento de Análisis Matemático
    E-18071 Granada, Spain
    e-mail

Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

Przepisz kod z obrazka

Odśwież obrazek

Odśwież obrazek