On the multiplicity of T\(\oplus T\oplus \cdot \cdot \cdot \oplus T\)
type: Item
instance of: scholarly article
title: On the multiplicity of T\(\oplus T\oplus \cdot \cdot \cdot \oplus T\)
zbMATH Open document ID: 0734.47012
author: Domingo A. Herrero
author: Warren R. Wogen
published in: Rocky Mountain Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: Let T be a bounded linear operator on a separable, complex, infinite dimensional Banach space X. The multiplicity of T is \(\mu (T)=\min card \Gamma\) where the minimum is taken over all \(\Gamma\subset X\) for which the closed linear span of \(\{T^ ky:\) \(y\in \Gamma\), \(k=0,1,2,...\}\) equals X. For \(n\geq 1\) let \(T^{(n)}\) denote the direct sum of n copies of T acting on \(X^{(n)}\), the direct sum of n copies of X and let \(T^{(\infty)}\), \(X^{(\infty)}\) extends this to countably infinite direct sums. The authors are interested in determining which sequences \(\{\mu_ n\}\) of natural numbers satisfy \(\mu_ n=\mu (T^{(n)})\) for some operator T. By combining some well-known and some new examples they are able to show that a number of sequences are attainable. It is not known whether others exist. They prove that \(\{\mu (T^{(\infty)})\}\) is either an unbounded sequence and then \(\mu (t^{(\infty)})=\infty\) or is bounded and \(\mu (T^{(\infty)})=\max \mu (T^{(n)})\).
Mathematics Subject Classification ID: 47A65
zbMATH DE Number: 4214758
zbMATH Keywords: direct sums
zbMATH Keywords: multiplicity
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181073120
OpenAlex ID: W2034460710
cites work: Q3689716
cites work: Multilinear mappings and estimates of multiplicity
cites work: Q4054260
cites work: Stable rank in holomorphic function algebras
cites work: Extension problems and stable rank in commutative Banach algebras
cites work: Shorter Notes: A Necessary Condition for Quasitriangularity
cites work: Q5615632
cites work: Q3961338
cites work: Q5662952
cites work: On multicyclic operators
cites work: Q3892784
cites work: On the Multiplicities of the Powers of a Banach Space Operator
cites work: Q4722689
cites work: Q3694095
cites work: Stable Rank of the Disc Algebra
cites work: Control subspaces of minimal dimension and root vectors
cites work: Q4116981
cites work: On cyclicity of commutants
DOI: 10.1216/RMJM/1181073120
description: scientific article; zbMATH DE number 4214758