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                <term xml:lang="en">invariant subspace problem</term>
                <term xml:lang="en">hyper invariant subspace problem</term>
                <term xml:lang="en">compact perturbations of normal operators</term>
                <term xml:lang="en">compact perturbations of diagonal operators</term>
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              <p>In this paper, a sufficient condition for the existence of hyper invariant subspace of compact perturbations of multiplication operators on some Banach spaces is presented. An interpretation of this result for compact perturbations of normal and diagonal operators on Hilbert space is also discussed. An improvement of a result of [FX12] for compact perturbations of diagonal operators is also obtained.</p>
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