We say that a Banach space X satisfies the “descent spectrum equality” (in short, DSE) whenever, for every bounded linear operator T on X, the descent spectrum of T as an operator coincides with the descent spectrum of T as an element of the algebra of all bounded linear operators on X. We prove that the DSE is fulfilled by ℓ1, all Hilbert spaces, and all Banach spaces which are not isomorphic to any of their proper quotients (so, in particular, by the hereditarily indecomposable Banach spaces [8]), but not by ℓ p , for 1 < p ≤ ∞ with p ≠ 2. Actually, a Banach space is not isomorphic to any of its proper quotients if and only if it is not isomorphic to any of its proper complemented subspaces and satisfies the DSE.
展开▼
机译：我们说，每当X上每个有界线性算子T的T的下降谱与作为元素的T的下降谱相符时，Banach空间X都满足“下降谱相等”（简称DSE）。 X上所有有界线性算子的代数。我们证明DSE由ℓ 1 ，所有Hilbert空间和所有Banach空间满足，这些空间与它们的任意商均不同构（因此，在特别是，对于1 ≤∞且p≠2，通过遗传不可分解的Banach空间[8]），而不是通过ℓ p 来实现。当且仅当它与其任何适当的补充子空间不是同构且满足DSE的商。
展开▼
