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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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