##
**A hereditarily indecomposable \(\mathcal L_{\infty}\)-space that solves the scalar-plus-compact problem.**
*(English)*
Zbl 1223.46007

The scalar-plus-compact problem asks whether there exists an infinite dimensional Banach space \(X\) such that every bounded linear operator \(T: X\to X\) has the form \(T=\lambda\,\text{Id}+K\) with a compact operator \(K\). Informally, one says that such a Banach space has “very few operators”. This long-standing difficult problem is solved in the positive in the paper at hand.

Hitherto, the strongest result in this direction was the theorem by W. T. Gowers and B. Maurey [“The unconditional basic sequence problem”, J. Am.Math.Soc.6, No. 4, 851–874 (1993; Zbl 0827.46008)] that the hereditarily indecomposable spaces constructed in their paper admit “few operators” in the sense that every bounded linear operator \(T: X\to X\) has the form \(T=\lambda\,\text{Id}+S\) with a strictly singular operator \(S\). It is known, though, that the Gowers-Maurey spaces definitely do not have very few operators.

The authors’ construction of a space \({\mathfrak X}_K\) with very few operators uses ideas from the Gowers-Maurey paper, but also another likewise highly nontrivial ingredient, namely the Bourgain-Delbaen construction of “exotic” \({\mathcal L}_\infty\)-spaces; cf.[J. Bourgain and F. Delbaen, “A class of special \({\mathcal L}_\infty\) spaces”, Acta Math.145, 155–176 (1980; Zbl 0466.46024)]. Indeed, the authors’ original intention was to “merely” produce a hereditarily indecomposable space whose dual is isomorphic to \(\ell_1\) and hence very far from this class. (Let us recall that a Banach space \(X\) is hereditarily indecomposable, HI for short, if whenever a closed subspace \(Y\subset X\) is decomposed into a direct sum \(Y=Y_1\oplus Y_2\), then \(Y_1\) or \(Y_2\) is finite dimensional. In particular, \(X\) does not contain an unconditional basic sequence.)

In order to obtain an HI predual of \(\ell_1\), the authors generalise the Bourgain-Delbaen construction in that they base their approach on mixed Tsirelson or Schlumprecht type spaces where originally \(\ell_p\)-spaces were used.

This alone does not guarantee that the resulting space has very few operators; indeed, recent work by M. Tarbard [“Hereditarily indecomposable, separable \({\mathcal{L}}_\infty\) spaces with \(\ell_1\) dual having few operators, but not very few operators,” available from arXiv:1011.4776] gives an example of an HI \({\mathcal L}^\infty\)-space with few, but not very few operators. The authors consider it a piece of serendipity that special features of the Bourgain-Delbaen method allow them to complete their construction of \({\mathfrak X}_K\). The construction itself is highly demanding and technical; suffice it to say that a key notion is, as in the case of the Gowers-Maurey spaces, that of a rapidly increasing sequence of which three important types are identified. The space \({\mathfrak X}_K\) is also shown to be saturated with reflexive subspaces having HI duals.

It is clear from the definition, respectively follows from known results, that a Banach space \(X\) with very few operators also has the following properties: the space of compact operators from \(X\) to \(X\) is complemented in the space of bounded operators \(L(X)\); if one assumes complex scalars, every bounded linear operator on \(X\) has an invariant subspace; the algebra \(L(X)\) is amenable; if \(X^*\) is separable, then \(L(X)\) is separable as well. The space \({\mathfrak X}_K\) – before long to be called Argyros-Haydon space – is also the first (separable) infinite dimensional example of a Banach space with any of the above properties.

As usual with nontrivial constructions of Banach spaces (or indeed any mathematical objects), one has to choose certain parameters and thus even obtains a whole family of examples. A by-product of the authors’ work is the following: There exist continuum many separable Banach spaces \(X_\alpha\) such that, for \(\alpha\neq \beta\), every bounded linear operator from \(X_\alpha\) to \(X_\beta\) is compact. These Banach spaces can be described as being “very incomparable”.

In the last section, the authors pose some open problems and comment on recent, as yet unpublished work. For example, they mention joint work with D. Freeman, E. Odell, Th. Raikoftsalis, Th. Schlumprecht and D. Zisimopoulou showing, in particular, that every separable reflexive space embeds into an \({\mathcal L}_\infty\)-space with very few operators.

Hitherto, the strongest result in this direction was the theorem by W. T. Gowers and B. Maurey [“The unconditional basic sequence problem”, J. Am.Math.Soc.6, No. 4, 851–874 (1993; Zbl 0827.46008)] that the hereditarily indecomposable spaces constructed in their paper admit “few operators” in the sense that every bounded linear operator \(T: X\to X\) has the form \(T=\lambda\,\text{Id}+S\) with a strictly singular operator \(S\). It is known, though, that the Gowers-Maurey spaces definitely do not have very few operators.

The authors’ construction of a space \({\mathfrak X}_K\) with very few operators uses ideas from the Gowers-Maurey paper, but also another likewise highly nontrivial ingredient, namely the Bourgain-Delbaen construction of “exotic” \({\mathcal L}_\infty\)-spaces; cf.[J. Bourgain and F. Delbaen, “A class of special \({\mathcal L}_\infty\) spaces”, Acta Math.145, 155–176 (1980; Zbl 0466.46024)]. Indeed, the authors’ original intention was to “merely” produce a hereditarily indecomposable space whose dual is isomorphic to \(\ell_1\) and hence very far from this class. (Let us recall that a Banach space \(X\) is hereditarily indecomposable, HI for short, if whenever a closed subspace \(Y\subset X\) is decomposed into a direct sum \(Y=Y_1\oplus Y_2\), then \(Y_1\) or \(Y_2\) is finite dimensional. In particular, \(X\) does not contain an unconditional basic sequence.)

In order to obtain an HI predual of \(\ell_1\), the authors generalise the Bourgain-Delbaen construction in that they base their approach on mixed Tsirelson or Schlumprecht type spaces where originally \(\ell_p\)-spaces were used.

This alone does not guarantee that the resulting space has very few operators; indeed, recent work by M. Tarbard [“Hereditarily indecomposable, separable \({\mathcal{L}}_\infty\) spaces with \(\ell_1\) dual having few operators, but not very few operators,” available from arXiv:1011.4776] gives an example of an HI \({\mathcal L}^\infty\)-space with few, but not very few operators. The authors consider it a piece of serendipity that special features of the Bourgain-Delbaen method allow them to complete their construction of \({\mathfrak X}_K\). The construction itself is highly demanding and technical; suffice it to say that a key notion is, as in the case of the Gowers-Maurey spaces, that of a rapidly increasing sequence of which three important types are identified. The space \({\mathfrak X}_K\) is also shown to be saturated with reflexive subspaces having HI duals.

It is clear from the definition, respectively follows from known results, that a Banach space \(X\) with very few operators also has the following properties: the space of compact operators from \(X\) to \(X\) is complemented in the space of bounded operators \(L(X)\); if one assumes complex scalars, every bounded linear operator on \(X\) has an invariant subspace; the algebra \(L(X)\) is amenable; if \(X^*\) is separable, then \(L(X)\) is separable as well. The space \({\mathfrak X}_K\) – before long to be called Argyros-Haydon space – is also the first (separable) infinite dimensional example of a Banach space with any of the above properties.

As usual with nontrivial constructions of Banach spaces (or indeed any mathematical objects), one has to choose certain parameters and thus even obtains a whole family of examples. A by-product of the authors’ work is the following: There exist continuum many separable Banach spaces \(X_\alpha\) such that, for \(\alpha\neq \beta\), every bounded linear operator from \(X_\alpha\) to \(X_\beta\) is compact. These Banach spaces can be described as being “very incomparable”.

In the last section, the authors pose some open problems and comment on recent, as yet unpublished work. For example, they mention joint work with D. Freeman, E. Odell, Th. Raikoftsalis, Th. Schlumprecht and D. Zisimopoulou showing, in particular, that every separable reflexive space embeds into an \({\mathcal L}_\infty\)-space with very few operators.

Reviewer: Dirk Werner (Berlin)

### MSC:

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

46B28 | Spaces of operators; tensor products; approximation properties |

47A15 | Invariant subspaces of linear operators |

### Keywords:

scalar-plus-compact problem; Bourgain-Delbaen construction; Gowers-Maurey spaces; HI spaces; hereditarily indecomposable spaces; spaces with few operators; spaces with very few operators; \({\mathcal L}_\infty\)-spaces; isomorphic \(\ell_1\)-preduals
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\textit{S. A. Argyros} and \textit{R. G. Haydon}, Acta Math. 206, No. 1, 1--54 (2011; Zbl 1223.46007)

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