Below is the text of the article "Frame properties of operator orbits" by Ole Christensen, DOI: 10.1002/mana.201800344. The actual image is at the bottom.
In what follows, we write $L^2 := L^2(\mathbb{T})$. By $L^2_+$ we denote the subspace of $L^2$ consisting of all $f \in L^2$ with vanishing negative Fourier coefficients, i.e., $$ L^2_+ := \left\{ f \in L^2 \left| \int_\mathbb{T} z^nf(z)d|z|=0 \text{ for } n \ge 1\right. \right\}, $$ where $d|z|$ indicates integration with respect to the normalized arc length measure.
My Question:
I can't understand why the defined set $L^2_+$ is not empty and what is normalized arc length measure $d|z|$ and how to calculate the integral.
The Original Image:
