English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Let $\mathcal{H}$ be a complex, separable Hilbert space and$\mathcal{B}(\mathcal{H})$ denote the algebra of all bounded linear operatorsacting on $\mathcal{H}$. Given a unitarily-invariant norm $\| \cdot \|_u$ on$\mathcal{B}(\mathcal{H})$ and two linear operators $A$ and $B$ in$\mathcal{B}(\mathcal{H})$, we shall say that $A$ and $B$ are\emph{polynomially isometric relative to} $\| \cdot \|_u$ if $\| p(A) \|_u = \|p(B) \|_u$ for all polynomials $p$. In this paper, we examine to what extent anoperator $A$ being polynomially isometric to a normal operator $N$ implies that$A$ is itself normal. More explicitly, we first show that if $\| \cdot \|_u$ isany unitarily-invariant norm on $\mathbb{M}_n(\mathbb{C})$, if $A, N \in\mathbb{M}_n(\mathbb{C})$ are polynomially isometric and $N$ is normal, then$A$ is normal. We then extend this result to the infinite-dimensional settingby showing that if $A, N \in \mathcal{B}(\mathcal{H})$ are polynomiallyisometric relative to the operator norm and $N$ is a normal operator whosespectrum neither disconnects the plane nor has interior, then $A$ is normal,while if the spectrum of $N$ is not of this form, then there always exists anon-normal operator $B$ such that $B$ and $N$ are polynomially isometric.Finally, we show that if $A$ and $N$ are compact operators with $N$ normal, andif $A$ and $N$ are polynomially isometric with respect to the $(c,p)$-normstudied by Chan, Li and Tu, then $A$ is again normal.

Operators which are polynomially isometric to a normal operator