Compact perturbations resulting in hereditarily polaroid operators. (English) Zbl 1485.47008

Duggal, B. P.

Bull. Belg. Math. Soc. - Simon Stevin 24, No. 3, 381-392 (2017).

Summary: A Banach space operator \(A\in B({\mathcal X})\) is polaroid, \(A\in(\mathcal P)\), if the isolated points of the spectrum \(\sigma(A)\) are poles of the operator; \(A\) is hereditarily polaroid, \(A\in(\mathcal {HP})\), if every restriction of \(A\) to a closed invariant subspace is polaroid. It is seen that operators \(A\in(\mathcal {HP})\) have SVEP – the single-valued extension property – on \(\Phi_{sf}(A)=\{\lambda: A-\lambda \text{ is semi Fredholm}\}\). Hence \(\Phi^+_{sf}(A)=\{\lambda\in\Phi_{sf}(A): \operatorname{ind}(A-\lambda)>0\}=\varnothing\) for operators \(A\in(\mathcal {HP})\), and a necessary and sufficient condition for the perturbation \(A+K\) of an operator \(A\in B({\mathcal X})\) by a compact operator \(K \in B({\mathcal X})\) to be hereditarily polaroid is that \(\Phi_{sf}^+(A)=\varnothing\). A sufficient condition for \(A\in B({\mathcal X})\) to have SVEP on \(\Phi_{sf}(A)\) is that its component \(\Omega_a(A)=\{\lambda\in\Phi_{sf}(A): \operatorname{ind}(A-\lambda)\leq 0\}\) is connected. We prove: If \(A\in B({\mathcal H})\) is a Hilbert space operator, then a necessary and sufficient condition for there to exist a compact operator \(K\in B({\mathcal H})\) such that \(A+K\in(\mathcal {HP})\) is that \(\Omega_a(A)\) is connected.

MSC:

47A10	Spectrum, resolvent
47A55	Perturbation theory of linear operators
47A53	(Semi-) Fredholm operators; index theories
47B40	Spectral operators, decomposable operators, well-bounded operators, etc.

Keywords:

Banach space operator; Hilbert space; compact perturbation; SVEP; hereditarily polaroid; Fredholmness