ON THE COMPLETENESS OF SOME-SPACES OFOPERATOR-VALUED FUNCTIONS
Lutz Klotz
Abstract: In [3] there were studied Banach spaces of (equivalence classes of) functions
whose values are unbounded operators, in general, and which are -integrable with respect
to operator-valued measures having an operator density with respect to some
non-negative scalar measure In the present short note it is shown that the values
of all functions are even bounded linear operators if and only if there is not
any set of positive finite measure such that the values of on have
non-closed ranges. The result is used to answer a question raised by Górniak et al. [2].
whose values are unbounded operators, in general, and which are 
-integrable with respect
to operator-valued measures having an operator density 
with respect to some
non-negative scalar measure 
In the present short note it is shown that the values
of all functions 
are even bounded linear operators if and only if there is not
any set 
of positive finite measure 
such that the values of 
on 
have
non-closed ranges. The result is used to answer a question raised by Górniak et al. [2].