Abstract
Extremal vectors were introduced by S. Ansari and P. Enflo in [2], this method produced new and more constructive proofs of existence of invariant subspaces. In this paper, our purpose is to introduce generalized extremal vectors and to study their properties. We firstly check that general properties of extremal vectors also hold for generalized extremal vectors. We give a new useful characterization of generalized extremal vectors. We show that there exist relationships between these vectors and the famous Moore–Penrose pseudo-inverse showing their intrinsic nature. Applications to weighted shift operators are given. In particular, we discuss for quasinilpotent backward weighted shifts the following question: Can the Ansari–Enflo method be used in order to obtain all hyper-invariant subspaces?
Mathematics Subject Classification (2010). Primary 47A15, 47B37; Secondary 47A50, 47S30.
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Cassier, G., Verliat, J. (2014). Generalized Extremal Vectors and Some New Properties. In: Cepedello Boiso, M., Hedenmalm, H., Kaashoek, M., Montes Rodríguez, A., Treil, S. (eds) Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation. Operator Theory: Advances and Applications, vol 236. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0648-0_7
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DOI: https://doi.org/10.1007/978-3-0348-0648-0_7
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