Spaces with a graph norm and strengthened Sobolev spaces. I (Q2386962)

The author investigates properties of Banach spaces with graph norm, i.e., spaces of the form \(\gamma(U,X)=\{ u\in U: \gamma u\in X\}\), where \(U\) and \(X\subset Y\) are Banach spaces and \(\gamma:U\rightarrow Y\) is a bounded linear operator. The norm is given by \(\| u\| _{\gamma(U,X)}=\| u\| _{U}+\| \gamma(u)\| _{X}\). Such properties like density of sets, compactness of embeddings and interpolation of operators are considered. The main examples are strengthened Sobolev spaces on bounded domains in which the norm is the sum of the usual Sobolev norm plus the norm of the trace on the boundary.