Extension of bounded linear operators.

*(English)*Zbl 1048.46016
Johnson, W. B. (ed.) et al., Handbook of the geometry of Banach spaces. Volume 2. Amsterdam: North-Holland (ISBN 0-444-51305-1/hbk). 1703-1741 (2003).

Let \(X\) be a Banach space and let \(E\) be a subspace of \(X\). By the Hahn-Banach theorem, every bounded linear functional given on \(E\) has a norm-preserving extension to the whole space \(X\). The Hahn-Banach theorem is the starting point of all extension theories which deal with more general operator extension problems. However, as the author points out, any attempt to generalize the Hahn-Banach theorem necessarily requires some restrictions: restrictions on the domain spaces \(X\) and \(E\), restrictions on the range space \(Y\), relaxation of the norm-preservation condition or restrictions on the class of operators to be extended.

The present survey article discusses in detail the main six extension problems, devoting to each of them one section of the article. Section 1 is the Introduction. It introduces each of the next sections in an easy-to-read manner providing a self-contained concise overview of the subject. Section 2 studies the injective Banach spaces. A Banach space \(E\) is called injective if it is complemented in every \(X\) containing \(E\). This is equivalent to the extension property of the pair \((E,X)\) for every \(X\) containing \(E\), meaning that, for every Banach space \(Y\), every operator \(T:E\rightarrow Y\) admits a bounded linear extension \(T:X\rightarrow Y\). It turns out that injective spaces are either finite-dimensional or non-separable. Separably injective spaces are discussed in Section 3 together with Rosenthal’s recent approach to separable injectivity in the case of non-separable spaces. This section is mainly devoted to the proof of the converse of Sobczyk’s theorem (1941) on the separable injectivity of \(c_0\): an infinite-dimensional separable Banach space which is complemented in every separable space containing it must be isomorphic to \(c_0\). This is a difficult result due to the author (1977) who has taken care to polish and simplify the original proof as much as possible (see, e.g., “Step 2. The construction of \(F\)”). Section 4 presents the extension theory for compact operators developed in Lindenstrauss’s Memoir (1964). Section 5 discusses lifting of operators and extension of into-isomorphisms to automorphisms of \(c_0\) and \(\ell_\infty\), and of operators from subspaces of \(\ell_1\). Extension of operators into \(C(K)\)-spaces is studied in Section 6 and an elegant proof of Maurey’s extension theorem is presented in Section 7.

This is an attractive survey article that gives a clear and lucid overview of the history and current knowledge of the area. It also contains many open problems with an inspiring discussion of them.

For the entire collection see [Zbl 1013.46001].

The present survey article discusses in detail the main six extension problems, devoting to each of them one section of the article. Section 1 is the Introduction. It introduces each of the next sections in an easy-to-read manner providing a self-contained concise overview of the subject. Section 2 studies the injective Banach spaces. A Banach space \(E\) is called injective if it is complemented in every \(X\) containing \(E\). This is equivalent to the extension property of the pair \((E,X)\) for every \(X\) containing \(E\), meaning that, for every Banach space \(Y\), every operator \(T:E\rightarrow Y\) admits a bounded linear extension \(T:X\rightarrow Y\). It turns out that injective spaces are either finite-dimensional or non-separable. Separably injective spaces are discussed in Section 3 together with Rosenthal’s recent approach to separable injectivity in the case of non-separable spaces. This section is mainly devoted to the proof of the converse of Sobczyk’s theorem (1941) on the separable injectivity of \(c_0\): an infinite-dimensional separable Banach space which is complemented in every separable space containing it must be isomorphic to \(c_0\). This is a difficult result due to the author (1977) who has taken care to polish and simplify the original proof as much as possible (see, e.g., “Step 2. The construction of \(F\)”). Section 4 presents the extension theory for compact operators developed in Lindenstrauss’s Memoir (1964). Section 5 discusses lifting of operators and extension of into-isomorphisms to automorphisms of \(c_0\) and \(\ell_\infty\), and of operators from subspaces of \(\ell_1\). Extension of operators into \(C(K)\)-spaces is studied in Section 6 and an elegant proof of Maurey’s extension theorem is presented in Section 7.

This is an attractive survey article that gives a clear and lucid overview of the history and current knowledge of the area. It also contains many open problems with an inspiring discussion of them.

For the entire collection see [Zbl 1013.46001].

Reviewer: Eve Oja (Tartu)

##### MSC:

46B03 | Isomorphic theory (including renorming) of Banach spaces |

46B04 | Isometric theory of Banach spaces |

46B20 | Geometry and structure of normed linear spaces |

46B25 | Classical Banach spaces in the general theory |

47A05 | General (adjoints, conjugates, products, inverses, domains, ranges, etc.) |

47A20 | Dilations, extensions, compressions of linear operators |

46-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to functional analysis |