Kolmogorov and Riesz type criteria of compactness in Köthe spaces of vector valued functions (Q750830)

Let G denote a locally compact abelian group, \({\mathcal B}(G)\) the \(\sigma\)- algebra of Borel subsets of G, \(\mu\) the Haar measure on \({\mathcal B}(G)\), and \({\mathcal B}_ 0(G)\) the family of Borel subsets of G with finite measure. Given a separable Banach space \((X,\| \cdot \|)\), let \({\mathcal M}(G,X)\) denote the family of all strongly measurable functions f: \(G\to X\). Let \(L^ 1_ X\subset {\mathcal M}(G,X)\) be the Lebesgue-Bochner space with the norm \(\| f\|_ 1=\int_{G}\| f(t)\| d\mu (t)\) and \(L^{\infty}_ X\subset {\mathcal M}(G,X)\) be the space of all essentially bounded strongly measurable functions from G to X with the norm \(\| f\|_{\infty}=\sup ess_{t\in G}\| f(t)\|\). For every \(A\in {\mathcal B}(G)\), let \(\chi_ A\) denote the characteristic function of A. A Banach space \((\Lambda_ X,\| \cdot \|_{\Lambda_ X})\subset {\mathcal M}(G,X)\) is said to be a Köthe space if it has the following properties: (a) \(\chi_ Af\in \Lambda_ X\) for all \(A\in {\mathcal B}(G)\) and \(f\in \Lambda_ X;\) (b) for all A,B\(\in {\mathcal B}(G)\) with \(A\subset B\) and for all \(f\in \Lambda_ X\), \(\| \chi_ Af\|_{\Lambda_ X}\leq \| \chi_ Bf\|_{\Lambda_ X};\) (c) for every \(A\in {\mathcal B}_ 0(G)\) the embeddings \(\chi_ AL^{\infty}_ X\hookrightarrow \chi_ A\Lambda_ X\hookrightarrow \chi_ AL'_ X\) are continuous with respect to the norms \(\| \cdot \|_{\infty}\), \(\| \cdot \|_{\Lambda_ X}\), and \(\| \cdot \|_ 1\), respectively. In this paper the authors study these Köthe spaces, and the results are applied to generalized Orlicz spaces of vector-valued functions. The problem of compactness of subsets in Köthe spaces is examined.