Maximal Ideals in Some Spaces of Bounded Linear Operators

Leung, Denny H.

doi:10.1017/s0013091517000219

Cited by 1 publication

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“…It is well known that the set of compact operators is the unique maximal ideal of when or [5]. In these cases, the set of compact operators coincides with the set There are many other Banach spaces X for which is the unique maximal ideal of , including [4], [3], , [13, 8, 10–12], [2], [16], [7] and an Orlicz sequence space which is close to [14].…”

Section: Introductionmentioning

confidence: 99%

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The Maximal Ideal in the Space of Operators On

CADAVID

Monika

Zheng

2022

Bull. Aust. Math. Soc.

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We study the isomorphic structure of $(\sum {\ell }_{q})_{c_{0}}\ (1< q<\infty )$ and prove that these spaces are complementably homogeneous. We also show that for any operator T from $(\sum {\ell }_{q})_{c_{0}}$ into ${\ell }_{q}$ , there is a subspace X of $(\sum {\ell }_{q})_{c_{0}}$ that is isometric to $(\sum {\ell }_{q})_{c_{0}}$ and the restriction of T on X has small norm. If T is a bounded linear operator on $(\sum {\ell }_{q})_{c_{0}}$ which is $(\sum {\ell }_{q})_{c_{0}}$ -strictly singular, then for any $\epsilon>0$ , there is a subspace X of $(\sum {\ell }_{q})_{c_{0}}$ which is isometric to $(\sum {\ell }_{q})_{c_{0}}$ with $\|T|_{X}\|<\epsilon $ . As an application, we show that the set of all $(\sum {\ell }_{q})_{c_{0}}$ -strictly singular operators on $(\sum {\ell }_{q})_{c_{0}}$ forms the unique maximal ideal of $\mathcal {L}((\sum {\ell }_{q})_{c_{0}})$ .

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Section: Introductionmentioning

confidence: 99%

“…There are many other Banach spaces X for which M X is the unique maximal ideal of L(X), including L p (0, 1) [13,8,[10][11][12], ( q ) p (1≤q< p<∞) [2], ( q ) 1 (1 < q < ∞) [16], d w,p [7] and an Orlicz sequence space which is close to p [14].…”

Section: Introductionmentioning

confidence: 99%