Banach space valued stochastic processes of weak second order on a locally compact abelian group
G
G
is considered. These processes are recognized as operator valued processes on
G
G
. More fully, letting
U
\mathfrak {U}
be a Banach space and
H
\mathfrak {H}
a Hilbert space, we study
B
(
U
,
H
)
B(\mathfrak {U},\mathfrak {H})
-valued processes. Since
B
(
U
,
H
)
B(\mathfrak {U},\mathfrak {H})
has a
B
(
U
,
U
∗
)
B(\mathfrak {U},\mathfrak {U}^*)
-valued gramian, every
B
(
U
,
H
)
B(\mathfrak {U},\mathfrak {H})
-valued process has a
B
(
U
,
U
∗
)
B(\mathfrak {U},\mathfrak {U}^*)
-valued covariance function. Using this property we can define operator stationarity, operator harmonizability and operator
V
V
-boundedness for
B
(
U
,
H
)
B(\mathfrak {U},\mathfrak {H})
-valued processes, in addition to scalar ones. Interrelations among these processes are obtained together with the operator stationary dilation.
Positive functionals and oscillation criteria for second order differential systems