1) takes values in a Banach space X (state space), the

unbounded operator (A,D(A)) generates a [C.

Unbounded Operator Algebras and Representation Theory.

Thus the bounded integral operator C (t) (D (t)) with |t| < 1 gives the right (left) inverse of the

unbounded operator A ([t.

If S: D(S) [subset] K [right arrow] L is another

unbounded operator, then ST: D(ST) [subset] H [right arrow] L is a new operator for which D(ST) = {x [member of] D(T): Tx [member of] D(S)} and (ST)x = S(Tx) for all x [member of] D(ST).

In [19] there is an example of a linear but

unbounded operator T which is hypercyclic but neither [T.

In what follows, we assume that {A(t), t [greater than or equal to] 0} is a family of closed densely defined linear

unbounded operators on the Banach space E and with domain D(A(t)) independent of t.

Triggiani: Extensions of rank conditions for controllability and observability to Banach space and

unbounded operators, SIAM J.

of Berlin) presents iterative splitting methods for decomposing evolution equations and decoupling differential equations with

unbounded operators.

She covers normed spaces and operators, Frechet spaces and Banach theorems, duality, weak topologies, distributions, the Fourier transform and Sobolev spaces, Banach algebras, and

unbounded operators in a Hilbert space.

The case with an unbounded non-selfadjoint operator as a main operator in a corresponding time invariant system has not been investigated sufficiently because of a number of essential difficulties usually related to

unbounded operators (especially non-selfadjoint ones).

Three appendices develop the theory of ultrapowers, discuss the basic theory of

unbounded operators, and provide an alternative approach from that of the main text to the existence of the trace in the basic construction.

The last result in this paper is a generalization of Theorem 3 (see below) to

unbounded operators.