On the Pointwise Bishop–Phelps–Bollobás Property for Operators

Abstract

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-Bollobás 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 Lp(µ) 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.arXiv:1709.00032v2 [math.FA] 26 Sep 2018 1 , Y ) fails the BPBp (even though all elements in L( 2 1 , Y ) attain their norms) [3]. Moreover, such Y can be found in a way that for every Banach space X, the set of all norm attaining operators is dense in L(X, Y ) [8, Theorem 4.2]. For more results and background on the BPBp, we refer the reader to the recent papers [1,6,9,10] and the references therein.In the very recent paper [13], the following stronger version of the Bishop-Phelps-Bollobás property was introduced (with the name of Bishop-Phelps-Bollobás point property).Definition 1.1 ([13, Definition 1.2]). We say that a pair (X, Y ) of Banach spaces has the pointwise Bishop-Phelps-Bollobás property (pointwise BPB property, for short) if given ε > 0, there existsη(ε) > 0 such that whenever T ∈ L(X, Y ) with T = 1 and x 0 ∈ S X satisfyThe difference between the BPBp and this new property is that, in the last one, the point x 0 does not move and the new operator S attains its norm at it. Also observe that, by an easy change of parameters, we may consider T ∈ L(X, Y ) with T 1 in the above definition, and we will use this fact without any explicit reference. Nevertheless, the same trick does not work for the point x 0 (as the operator S attains its norm at x 0 ), so we have to consider x 0 ∈ S X .Another useful definitions, motivated by the corresponding ones for the BPBp given in [8], are the following.Definition 1.2. A Banach space X is said to be a universal pointwise BPB domain space if (X, Y ) has the pointwise BPB property for every Banach space Y . A Banach space Y is said to be a universal pointwise BPB range space if (X, Y ) has the pointwise BPB property for every uniformly smooth Banach space X.As we already mentioned, the pointwise BPB property was introduced in [13] where, among other results, it was proved that if (X, Y ) has the pointwise BPB property then X must be uniformly smooth, that Hilbert spaces are universal pointwise BPB domain spaces, and that the class of universal pointwise BPB range spaces includes uni...

Strong subdifferentiability and local Bishop–Phelps–Bollobás properties

“…Also, it was proved in [18] that the pair (X; K) has the BPBpp if and only if X is uniformly smooth. In both papers [18,19] the authors presented such differences between these two properties and found many positive examples having BPBpp.…”

Local Bishop–Phelps–Bollobás properties

“…Many different variants of the Bishop-Phelps-Bollobás theorem were introduced during the last years. For some of them, we refer the recent papers [12,13,14,15]. Our aim is to study local versions of these properties, as in [16].…”

“…When x 0 = x in the previous definition, we say that (X, Y ) has the Bishop-Phelps-Bollobás point property (BPBpp, for short); this property was defined and studied in [13,14]. If instead of fixing the point x (as in the BPBpp) we fix the operator T , we say that (X, Y ) has the Bishop-Phelps-Bollobás operator property (see [12,15]).…”

The Bishop-Phelps-Bollobás properties in complex Hilbert spaces

“…Recently, a stronger property than the BPBp was introduced in [13], which is called the Bishop-Phelps-Bollobás point property (or the pointwise Bishop-Phelps-Bollobás property [11,12]). We say that the pair (X, Y ) satisfies the Bishop-Phelps-Bollobás point property (BPBpp, for short) if given ε > 0, there is η(ε) > 0 such that whenever T ∈ L(X, Y ) with T = 1 and x 0 ∈ S X satisfy…”

“…For this reason, the BPBpp implies the BPBp, but the converse is not true in general (see [13,Proposition 2.3]): if a pair (X, Y ) satisfies the BPBpp, then the domain space X must be uniformly smooth. In both papers [11,13], the authors extended some known results about the BPBp to this new property, and presented a pair (X, Y ) to fail the BPBpp for some uniformly smooth Banach spaces X. In particular, they showed that if H is a Hilbert space, then the pair (H, Y ) always has the BPBpp for every Banach space Y (see also [12] for a more general result).…”