Data: wtorek, 15.03.2022, godz. 10:30-12:00
Prelegent: dr Tomasz Kobos (Instytut Matematyki UJ)
Abstrakt: If \(X\) is a normed space, then a semi-inner product on \(X\) is a mapping, which satisfies the same properties as inner product, but it is not necessarily symmetric/conjugate-symmetric. This notion was first introduced by Lumer and then later redefined by Giles. The main motivation behind it, was to transfer certain Hilbert space arguments to the setting of general Banach spaces, as for every normed space there exists a semi-inner product generating its norm. Moreover, a semi-inner product is uniquely determined if and only if \(X\) is a smooth space.
One of the fundamental properties of an inner product says that if \(X\) is an inner product space and \(f\colon X \to X\) is a mapping that preserves the inner product, then f has to be a linear isometry. Our goal is to investigate a similar question, but for mappings preserving the semi-inner product in smooth normed spaces.
We say that a smooth normed space \(X\) has a property (SL), if every mapping \(f\colon X \to X\) preserving the semi-inner product on \(X\) is linear. It is well known that every Hilbert space has the property (SL) and the same is true for every finite-dimensional smooth normed space. Our aim is to prove several new results concerning the property (SL). We give a simple example of a smooth and strictly convex Banach space which is isomorphic to the space \(l_p\), but without the property (SL). Moreover, we provide a characterization of the property (SL) in the class of reflexive smooth Banach spaces in terms of subspaces of quotient spaces. As a consequence, we prove that the space \(l_p\) have the property (SL) for every \(1 < p < \infty\). Finally, using a variant of the Gowers-Maurey space, we construct a uniformly smooth Banach space \(X\) such that every smooth Banach space isomorphic to \(X\) has the property (SL).