# Resolvent criteria for similarity to a normal operator with spectrum on a curve

## TL;DR

This paper introduces new resolvent-based criteria for determining when a Hilbert space operator with spectrum on a smooth curve is similar to a normal operator, extending previous results and addressing open questions.

## Contribution

It generalizes existing similarity criteria by employing resolvent estimates and dilation techniques, advancing the understanding of operator similarity to normal operators.

## Key findings

- New resolvent criteria for similarity to normal operators.
- Extension of previous criteria by Stampfli, Van Casteren, and Naboko.
- Application of dilation and Dynkin functional calculus methods.

## Abstract

We give some new criteria for a Hilbert space operator with spectrum on a smooth curve to be similar to a normal operator, in terms of pointwise and integral estimates of the resolvent. These results generalize criteria of Stampfli, Van Casteren and Naboko, and answer several questions posed by Stampfli. The main tools are from our recent results on dilation to the boundary of the spectrum, along with the Dynkin functional calculus for smooth functions, which is based on pseudoanalytic continuation.