A Gowers tree like space and the space of its bounded linear operators

Petsoulas, Giorgos; Raikoftsalis, Theocharis

doi:10.4064/sm190-3-2

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2013

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“…The structure of L(X) has been also studied for non-reflexive HI spaces ( [13], [4], [27]). It is notable that in all these examples, each strictly singular operator T ∈ L(X) is a weakly compact one.…”

Section: Introductionmentioning

confidence: 99%

Strictly singular non-compact diagonal operators on HI spaces

Argyros,

Deliyanni,

Tolias

2008

Preprint

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We construct a Hereditarily Indecomposable Banach space X d with a Schauder basis (en) n∈N on which there exist strictly singular noncompact diagonal operators. Moreover, the space L diag (X d ) of diagonal operators with respect to the basis (en) n∈N contains an isomorphic copy of ℓ∞(N). Contents 1. Introduction 1 2. The mixed Tsirelson space T 0 = T [(A nj , 1 mj ) ∞ j=1 ] 6 3. Strictly singular non-compact operators on T 0 14 4. The HI space X d 17 5. A class of bounded diagonal operators on X d 26 6. The structure of the space L diag (X d ) 31 7. The finite block representability of J T0 in X d 33 References 45 2000 Mathematics Subject Classification. 46B28, 46B20, 46B03.

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Section: Introductionmentioning

confidence: 99%