On the convergence of the normalized power sequence of Riesz operators

Abstract

Let $\mathscr{H}$ be a complex Hilbert space and $\mathscr{B}(\mathscr{H})$ be the algebra of all bounded linear operators on $\mathscr{H}$. For $T \in \mathscr{B}(\mathscr{H})$, let $|T| := (T^*T)^{\frac{1}{2}}$. We refer to the sequence $\big\{|T^n|^{\frac{1}{n}}\big\}_{n \in \mathbb{N}}$ as the NPS (normalized power sequence) of $T$. In this article, we show that the NPS of a Riesz operator $R \in \mathscr{B}(\mathscr{H})$ converges in norm to a positive operator $H$, and provide an explicit description of the spectral resolution of $H$ in terms of the Riesz idempotents associated with the non-zero eigenvalues of $R$. Since every compact operator is a Riesz operator, this gives us a stronger, spatial generalization of the Yamamoto-Davis theorem, which asserts that $\lim_{n \to \infty} s_j(K^n)^{\frac{1}{n}}$ is equal to the $j^{\textrm{th}}$-largest eigenvalue-modulus of the compact operator $K$, where $s_j(\cdot)$ denotes the $j^{\textrm{th}}$-largest singular value. In recent work, the present authors have established the norm convergence of the NPS for spectral operators. Using a rank-one perturbation of a unitary operator, we demonstrate that this fails in general for essentially spectral operators (of which Riesz operators form a subclass).