×

Optimal domain and integral extension of operators, acting in function spaces. (English) Zbl 1145.47027

Operator Theory: Advances and Applications 180. Basel: Birkhäuser (ISBN 978-3-7643-8647-4/hbk). xii, 400 p. (2008).
This book deals with the contemporary theme of extending integral operators to an optimal domain. The authors are among the main contributors to this area and have made the effort to give a very comprehensive view of this area of Analysis.
In Chapter 1, the authors introduce various notations, terminology and also give a very cogent idea of what is to be expected in the later chapters. To make the monograph self-contained, an exposition on quasi-Banach function spaces (q-B.f.s), vector measures and integration operators is covered in Chapters 2 and 3.
Let \(X(\mu)\) be a \(\sigma\)-order continuous q-B.f.s and let \(E\) be a Banach space. Let \(T: X(\mu) \rightarrow E\) be a continuous linear operator. Consider the associated vector measure \(m_T(A)=T(\chi_A)\). Then \(T\) is said to be \(\mu\)-determined if \({\mathcal N}(\mu) = {\mathcal N}(m_T)\). In this case, \(X(\mu)\) is continuously embedded in \(L^1(m_T)\) and \(L^1(m_T)\) is the largest space (on the given finite measure space) among q-B.f.s’s with \(\sigma\)-a.c quasi-norm into which \(X(\mu)\) is continuously embedded, and to which \(T\) admits an \(E\)-valued continuous linear extension. Such an extension is unique and is precisely the integral operator \(I_{m_T}\). Chapter 4 contains several concrete example from analysis that illustrate this phenomenon.
Chapter 5 deals with \(\mu\)-determined \(p\)-th power operators (\(1\leq p<\infty\)). Here one considers \(X(\mu)_{[p]}\), the \(p\)-th power of \(X(\mu)\), equipped with the quasi-norm \(\|| f|^{\frac{1}{p}}\|^p_{X(\mu)}\). The inclusion map is denoted by \(i_{[p]}\) and \(T\) is \(p\)-th power factorable if \(T=T_{[p]}\circ i_{[p]}\) for some continuous linear operator \(T_{[p]}: X(\mu)_{[p]}\rightarrow E\). It is said to admit an \({\mathcal F}_{[p]}\)-extension to a B.f.s \(Y(\mu) \supset X(\mu)\), if there is a \(\mu\)-determined \(p\)th-power factorable extension operator. It turns out that the B.f.s \(L^p(m_T)\) is the largest one within the class of all \(\sigma\)-order continuous B.f.s for which \(T\) can be extended.
Analogous to the Maurey–Rosenthal factorization, in Chapter 6 the authors consider factorization of \(p\)-th power factorable operators through \(L^q\)-spaces. Again, for a \(\mu\)-determined \(T\in {\mathcal L}(X(\mu),E)\), conditions are given under which it is bidual \((p,q)\)-power concave. In particular, when \(X(\mu)\) is both \(q\)-convex and \(q\)-concave, this is equivalent to \(p\)-th power factorability.
For a compact abelian group \(G\), the study of Banach space theoretic properties of various classical operators on \(L^p(G)\) forms the content of Chapter 7. For example, for \(1<p \leq \infty\), the operator \(F_1:L^1(G) \rightarrow \ell^{\infty}(\Gamma)\) (\(\Gamma\) denotes the dual group) is not weakly compact, whereas \(F_p: L^1(G) \rightarrow \ell^p(\Gamma)\) is weakly compact. Let \(M_0(G)=\{\lambda \in M(G): \lambda^{\wedge} \in c_0(\Gamma)\}\). There are 15 operator theoretic characterizations of this set. For example, \(\lambda \in M_0(G)\) if and only if the convolution operator \(C^{(1)}_{\lambda} : L^1(G) \rightarrow L^1(G)\) is completely continuous.
As mentioned on the book cover, many of the topics considered here are in their infancy and it is hoped that this well motivated monograph will steer several young researchers towards this active area.

MSC:

47B38 Linear operators on function spaces (general)
28B05 Vector-valued set functions, measures and integrals
28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46G10 Vector-valued measures and integration
47B07 Linear operators defined by compactness properties
47B34 Kernel operators
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
PDFBibTeX XMLCite