For a bounded linear operator $A$ on a complex Hilbert space $\mathcal{H}$, the Douglas-Yang metric on the resolvent set $\rho(A)$ is defined by the metric function $g_{\vec{x}}(z)=\left \| \big(A -z I\big)^{-1} \vec{x} \right \|^2$, where $\vec{x} \in \mathcal{H}$ with $\left \| \vec{x} \right \|=1$.

The Douglas-Yang metric provide the opportunity to see what particular properties relating to the operator $A$ can be determine by the use of geometry. We look at how the arc length, curvature, and geodesic equations are defined with respect to the Douglas-Yang metric in the case when $A$ is nilpotent operator and the unilateral shift operator. When $A$ is the unilateral shift operator on the Hardy space $H^2(\mathbb{D})$, the metric function $g_{f}(z)$ with respect to the Douglas-Yang metric is dependent on the chosen function $f \in H(\mathbb{D})$. We determine the extremal values of the arc length of the circle $C_{r}$ centered at the origin with radius $r$ with respect to the choice of $f$ in the Douglas-Yang metric. We showed that the maximal value is reached precisely when $f$ is an inner function.

Other considerations include defining the metric function $f_{A}(z)$ by using trace, extremal values of the Ricci curvature with respect to the Douglas-Yang metric. And it appears that its extremal value is related to the normalized trace of the matrix $A$.