0]-semigroup S (t), t [greater than or equal to] 0, on E, F is a strongly measurable (in the strong operator topology) and uniformly bounded operator
valued function with values in L(E) and B is also a strongly measurable uniformly bounded operator
valued function with values in L(U, E) where U is any separable Hilbert space (so a Polish space) and [sigma] is a strongly measurable operator valued function taking values in [L.
0]) < 0) and a constant bounded operator
E has a solution Y which is represented as
t]f(x) is the mean value of f over the sphere of radius t centered at x and it defines a bounded operator
A kernel [phi]: G x G [right arrow] C is a Schur multiplier if for every bounded operator
A = [([a.
0] induces a bounded operator
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on [L.
Let A be a fixed linear bounded operator
acting on the Banach space [C.
ran(C)] is a bounded operator
from ran(C) onto ran(C) with bounded inverse.
An alternative proof of Theorem 10 can be given using the fact that, for X an arbitrary bounded operator
A linear bounded operator
on A is called a multiplier if it satisfies xT(y) = T(xy) for all x, y [member of] A.
It is well known that A is a bounded operator
on the Banach space m of bounded sequences if and only if
Notice that, in the case that A is a bounded operator
, the left-definite theory is trivial but, when A is unbounded, the theory has substance.
For any bounded operator
T : H [right arrow] H its Berezin symbol (see, Nordgren and Rosenthal  and Zhu ) is defined by