The Bishop-Phelps-Bollobás Theorem for bilinear forms (Q2849035)

Motivated by a quantitative version of the Bishop-Phelps theorem given by \textit{B. Bollobás} [Bull. Lond. Math. Soc. 2, 181--182 (1970; Zbl 0217.45104)], \textit{M. D. Acosta} et al. [J. Funct. Anal. 254, No. 11, 2780--2799 (2008; Zbl 1152.46006)] defined the so-called Bishop-Phelps-Bollobás property as a quantitative way in which the set of norm attaining operators can be dense in the space of all operators between two Banach spaces. Later on, \textit{Y. S. Choi} and \textit{H. G. Song} [J. Math. Anal. Appl. 360, No. 2, 752--753 (2009; Zbl 1190.46012)] introduced the corresponding property for bilinear forms. A pair of Banach spaces \((X,Y)\) has the Bishop-Phelps-Bollobás property for bilinear forms if, for every \(\varepsilon>0\), there exists \(\eta>0\) such that, whenever \(A\in L^2(X,Y)\) with \(\|A\|=1\), \(x_0\in X\) with \(\|x_0\|=1\) and \(y_0\in Y\) with \(\|y_0\|=1\) satisfy \(\|A(x_0,y_0)\|> 1 - \eta\), there exist \(x\in X\) with \(\|x\|=1\), \(y\in Y\) with \(\|y\|=1\), and \(B\in L^2(X,Y)\) with \(\|B\|=1\) such that NEWLINE\[NEWLINE \|B(x,y)\|=1,\quad \|x_0-x\|<\varepsilon,\quad \|y_0-y\|<\varepsilon,\quad \|T-S\|<\varepsilon. NEWLINE\]NEWLINE (\(L^2(X,Y)\) represents the space of continuous bilinear forms on \(X\times Y\).)NEWLINENEWLINEIn the paper under review, positive and negative results about this property are given. With respect to positive results, the authors show that a pair \((X,Y)\) has the Bishop-Phelps-Bollobás property for bilinear forms provided that \(X\) is uniformly convex (\(Y\) can be any Banach space). Next, the authors characterize those Banach spaces \(Y\) such that the pair \((\ell_1,Y)\) has the Bishop-Phelps-Bollobás property for bilinear forms in terms of convex series. This characterization provides that the pairs \((\ell_1,Y)\) have the Bishop-Phelps-Bollobás property for bilinear forms for finite-dimensional \(Y\), \(Y=C(K)\), and \(Y\) equals to the space of compact operators on a Hilbert space. On the other hand, it is shown that the pair \((\ell_1,L_1(\mu))\) fails the Bishop-Phelps-Bollobás property for bilinear forms for every infinite-dimensional \(L_1(\mu)\) space, a result that was known only when \(L_1(\mu)=\ell_1\) (see [Zbl 1190.46012]).