Weak supercyclicity: dynamics of paranormal operators

  • B. P. Duggal


If A is a Banach space operator with the single-valued extension property such that the isolated points of the spectrum of A are poles of (the resolvent of) A, then A is neither weakly hypercyclic nor n-supercyclic for any positive integer \(n\ge 1\). Furthermore, if also A and \(A^{-1}\) (whenever it exists) are normaloid, then A weakly supercyclic implies A is an invertible isometry. Translated to classes of Hilbert space operators, this implies that if an operator A is either paranormal or \(*\)-paranormal (or even M-hyponormal), then it is not weakly hypercyclic or n-supercyclic; furthermore, A weakly supercyclic implies A is a scalar multiple of a unitary operator [thereby answering a problem posed in Duggal et al. (J Math Anal Appl 427:107–113, 2015)].


Banach space operator Weak hypercyclicity n-supercyclicity Weak supercyclicity SVEP Polaroid operator Totally hereditarily normaloid operator Paranormal operator 

Mathematics Subject Classification

Primary 47B20 47A16 


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Copyright information

© Springer-Verlag Italia 2016

Authors and Affiliations

  1. 1.LondonUK

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