| Abstract: | In this paper an analog of the Blum-Hanson theorem for quantum quadratic processes on the von Neumann algebra is proved, i.e.,
it is established that the following conditions are equivalent:
| i) |
P(
t
)x is weakly convergent tox
0;
|
| ii) |
for any sequence {a
n} of nonnegative integrable functions on 1, ∞) such that ∝
1
∞
a
n(t)dt=1 for anyn and lim
n→∞ ∥a
n∥∞=0, the integral ∝
1
∞
a
n(t)P(
t
)x dt is strongly convergent tox
0 inL
2(M, ϕ), wherex ɛM,P(
t
) is a quantum quadratic process,M is a von Neumann algebra, andϕ is an exact normal state onM.
|
Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 102–109, January, 2000. |
