Bishop-Phelps-Bollob\'as property for bilinear forms on spaces of continuous functions

It is shown that the Bishop-Phelps-Bollob\'as theorem holds for bilinear forms on the complex $C_0(L_1)\times C_0(L_2)$ for arbitrary locally compact topological Hausdorff spaces $L_1$ and $L_2$

On the pointwise Bishop--Phelps--Bollob\'as property for operators

We study approximation of operators between Banach spaces $X$ and $Y$ that nearly attain their norms in a given point by operators that attain their norms at the same point. When such approximations exist, we say that the pair $(X, Y)$ has the pointwise Bishop-Phelps-Bollob\'as property (pointwise BPB property for short). In this paper we mostly concentrate on those $X$, called universal pointwise BPB domain spaces, such that $(X, Y)$ possesses pointwise BPB property for every $Y$, and on those $Y$, called universal pointwise BPB range spaces, such that $(X, Y)$ enjoys pointwise BPB property for every uniformly smooth $X$. We show that every universal pointwise BPB domain space is uniformly convex and that $L_p(\mu)$ spaces fail to have this property when $p>2$. For universal pointwise BPB range space, we show that every simultaneously uniformly convex and uniformly smooth Banach space fails it if its dimension is greater than one. We also discuss a version of the pointwise BPB property for compact operators.Comment: 19 pages, to appear in the Canadian J. Math. In this version, section 6 and the appendix of the previous version have been remove

The Bishop-Phelps-Bollob{\'a}s property for numerical radius of operators on L1(μ)L_1 (\mu)

In this paper, we introduce the notion of the Bishop-Phelps-Bollob\'as property for numerical radius (BPBp-$\nu$) for a subclass of the space of bounded linear operators. Then, we show that certain subspaces of $\mathcal{L}(L_1(\mu))$ have the BPBp-$\nu$ for every finite measure $\mu$. As a consequence we deduce that the subspaces of finite-rank operators, compact operators and weakly compact operators on $L_1(\mu)$ have the BPBp-$\nu$.Comment: 15 page

The version for compact operators of Lindenstrauss properties A and B

It has been very recently discovered that there are compact linear operators between Banach spaces which cannot be approximated by norm attaining operators. The aim of this expository paper is to give an overview of those examples and also of sufficient conditions ensuring that compact linear operators can be approximated by norm attaining operators. To do so, we introduce the analogues for compact operators of Lindenstrauss properties A and B.Comment: RACSAM (to appear). The final publication is available at Springer via http://dx.doi.org/10.1007/s13398-015-0219-

Bounded holomorphic functions attaining their norms in the bidual

Under certain hypotheses on the Banach space $X$, we prove that the set of analytic functions in $\mathcal{A}_u(X)$ (the algebra of all holomorphic and uniformly continuous functions in the ball of $X$) whose Aron-Berner extensions attain their norms, is dense in $\mathcal{A}_u(X)$. The result holds also for functions with values in a dual space or in a Banach space with the so-called property $(\beta)$. For this, we establish first a Lindenstrauss type theorem for continuous polynomials. We also present some counterexamples for the Bishop-Phelps theorem in the analytic and polynomial cases where our results apply.Comment: Accepted in Publ. Res. Inst. Math. Sc

For maximally monotone linear relations, dense type, negative-infimum type, and Fitzpatrick-Phelps type all coincide with monotonicity of the adjoint

It is shown that, for maximally monotone linear relations defined on a general Banach space, the monotonicities of dense type, of negative-infimum type, and of Fitzpatrick-Phelps type are the same and equivalent to monotonicity of the adjoint. This result also provides affirmative answers to two problems: one posed by Phelps and Simons, and the other by Simons.Comment: 15 page

On the Bishop-Phelps-Bollobas property for numerical radius in C(K)-spaces

We study the Bishop-Phelps-Bollobas property for numerical radius within the framework of C(K) spaces. We present several sufficient conditions on a compact space K ensuring that C(K) has the Bishop-Phelps-Bollobas property for numerical radius. In particular, we show that C(K) has such property whenever K is metrizable

Norm attaining operators of finite rank

We provide sufficient conditions on a Banach space $X$ in order that there exist norm attaining operators of rank at least two from $X$ into any Banach space of dimension at least two. For example, a rather weak such condition is the existence of a non-trivial cone consisting of norm attaining functionals on $X$. We go on to discuss density of norm attaining operators of finite rank among all operators of finite rank, which holds for instance when there is a dense linear subspace consisting of norm attaining functionals on $X$. In particular, we consider the case of Hilbert space valued operators where we obtain a complete characterization of these properties. In the final section we offer a candidate for a counterexample to the complex Bishop-Phelps theorem on $c_0$, the first such counterexample on a certain complex Banach space being due to V. Lomonosov.Comment: 25 pages, minor modifications, to appear in the special volume "The mathematical legacy of Victor Lomonosov", to be published by De Gruyte

A non-linear Bishop-Phelps-Bollob\'as type theorem

The main aim of this paper is to prove a Bishop-Phelps-Bollob\'as type theorem on the unital uniform algebra A_{w^*u}(B_{X^*}) consisting of all w^*-uniformly continuous functions on the closed unit ball B_{X^*} which are holomorphic on the interior of B_{X^*}. We show that this result holds for A_{w^*u}(B_{X^*}) if X^* is uniformly convex or X^* is the uniformly complex convex dual space of an order continuous absolute normed space. The vector-valued case is also studied

Symmetric multilinear forms on Hilbert spaces: where do they attain their norm?

We characterize the sets of norm one vectors $\mathbf{x}_1,\ldots,\mathbf{x}_k$ in a Hilbert space $\mathcal H$ such that there exists a $k$-linear symmetric form attaining its norm at $(\textbf{x}_1,\ldots,\mathbf{x}_k)$. We prove that in the bilinear case, any two vectors satisfy this property. However, for $k\ge 3$ only collinear vectors satisfy this property in the complex case, while in the real case this is equivalent to $\mathbf{x}_1,\ldots,\mathbf{x}_k$ spanning a subspace of dimension at most 2. We use these results to obtain some applications to symmetric multilinear forms, symmetric tensor products and the exposed points of the unit ball of $\mathcal L_s(^k\mathcal{H})$.Comment: 17 page