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 in [L.

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 [10] and Zhu [15]) is defined by