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Operators which do not have the single valued extension property. (English) Zbl 0978.47002
A bounded linear operator $$T$$ on a complex Banach space $$X$$ is said to have the single valued extension property (SVEP) at $$\lambda_0\in\mathbb{C}$$ if for every open disc $$D$$ centered at $$\lambda_0$$ the only analytic function $$f: D\to X$$ satisfying $$(\lambda I-T)f(\lambda)= 0$$ on $$D$$ is the constant function $$f= 0$$. $$T$$ is said to have the SVEP if it has the SVEP at every $$\lambda$$ in $$\mathbb{C}$$. The authors start by proving that $$T$$ does not have the SVEP at $$0$$ if and only if there is a nonzero vector $$x$$ in $$\ker T$$ such that the local spectrum of $$T$$ at $$x$$ is empty. This is a local version of the known result for $$T$$ not to have the SVEP. Then they proceed to relate this property to the ascent of $$T$$, which is the smallest positive integer $$p$$ such that $$\ker T^p= \ker T^{p+1}$$. It is proved that if $$T$$ does not have the SVEP at $$0$$, then its ascent is infinity. For a semi-Fredholm operator $$T$$, these two conditions are equivalent and are also equivalent to $$0$$ being a limit point of the eigenvalues of $$T$$. A consequence of this is that if $$T$$ has the SVEP, then its Browder spectrum and Weyl spectrum are equal. They also investigate many other properties and show how these are related to the SVEP.

##### MSC:
 47A11 Local spectral properties of linear operators 47A53 (Semi-) Fredholm operators; index theories 47A20 Dilations, extensions, compressions of linear operators
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