Geodesic
Isometrically equivalent composition operators on spaces of analytic vector-valued functions
Glasgow mathematical journal, Tome 41 (1999) no. 3, pp. 441-451

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Let $X$ be a Banach space and let $B(X)$ denote the space of bounded operators on $X$. Two elements $S,T\inB(X)$ are isometrically equivalent if there exists an invertible isometry $V$ such that $TV=VS$. If $X$ is a Hilbert space, then $V$ is a unitary operator and $S$ and $T$ are said to be unitarily equivalent. 
Hornor, William E.; Jamison, James E. Isometrically equivalent composition operators on spaces of analytic vector-valued functions. Glasgow mathematical journal, Tome 41 (1999) no. 3, pp. 441-451. doi: 10.1017/S0017089599000038
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