On the multiplicity of T\(\oplus T\oplus \cdot \cdot \cdot \oplus T\) (Q810844)

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 \(\mu (T^{(\infty)})=\max \mu (T^{(n)})\).