Geodesic
Measures on spaces of operators and isometries
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XV, Tome 149 (1986), pp. 127-136

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Let $\mu$ be a finite Borel (in the strong operator topology) measure on the space $B(E, F)$ of bounded linear operator from $E$ into $F$; $E$, $F$ being Banach spaces. Suppose that either $E=C(K)$, $F$ arbitrary, $p>1$ or $E=F=L^q(Y)$, $p>1$, $q>1$, $q\not\in[p,2]$. Suppose next that $\|e\|^p=\int\|Te\|^p\,d\mu(T)$ for every $e\in E$. Then $\mu$ is supported on scalar multiples of isometries. 
@article{ZNSL_1986_149_a10,
     author = {A. L. Koldobskii},
     title = {Measures on spaces of operators and isometries},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {127--136},
     publisher = {mathdoc},
     volume = {149},
     year = {1986},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1986_149_a10/}
}
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A. L. Koldobskii. Measures on spaces of operators and isometries. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XV, Tome 149 (1986), pp. 127-136. http://geodesic.mathdoc.fr/item/ZNSL_1986_149_a10/