In Theorem 14, we shall show that for a subset F of [C.sub.[phi]] if there exists a closed operator A in H such that T [subset] A for all T [member of] F, then F has a maximal element, and furthermore, if there exists a closed operator B in H such that [([T.sup.-1]).sup.*] [subset] B for all T [member of] F, then F have a maximal element and a minimal element.

Since D([T.sup.-1,sub.e,[phi]]) = [T.sub.e,[phi]] D ([T.sub.e,[phi]]) [contains] [T.sub.e,[phi]] [D.sub.[psi]] = [D.sub.e], [T.sup.-1.sub.e,[phi]] is a densely defined closed operator in H, and since D([T.sub.e,[phi]]) = D([phi]), it has a densely defined inverse [T.sub.e,[phi]].

is called a Hamiltonian operator matrix, if A is a closed operator with dense domain and B, C are self-adjoint operators.

be closed operator matrix such that A [member of] C(H), D [member of] C(K) with dense domains and let B [member of] [C.sup.+.sub.D] (K, H) = {B [member of] [C.sup.+] (K, H) : D(D) [subset] D(B), D [member of] C(K)}; then there exists some B [member of] [C.sup.+.sub.D] (K, H) such that T is upper semi-Weyl operator if and only if A is upper semi-Fredholm operator and

After then, Hirasawa and Miura [5] gave some necessary and sufficient conditions under which a

closed operator in a Hilbert space has the Hyers-Ulam stability.

Then, such a limit is a closed operator which is bounded if all the operators of the sequence are bounded.

(2) If the bounded sequence {[x.sup.(n).sub.m]} [subset] Dom([T.sub.n]) converges to [x.sup.(n)] [subset] Dom([T.sub.n]) for any n [member of] [N.sub.0] then {[T.sub.n][x.sup.(n).sub.m]} [right arrow] [T.sub.n][x.sup.(n)] for any n [member of] [N.sub.0] as m [right arrow] [infinity] since {[T.sub.n]} is a closed operator for any n [member of] [N.sub.0] so that

Then Weaver goes on to define a quantum relation on a von Neumann algebra to be a weak*

closed operator bimodule over its commutant.

We can define the order

closed operator on ordered dualistic partial metric spaces by the following way.

The synthesis operator of an algebraic frame generated by the unbounded operator T: D(T) [subset] H [right arrow] [L.sup.2]([mu]), must be a densely defined

closed operator S: D(S) [subset] [L.sup.2]([mu]) [right arrow] H acting as a left inverse to T.

This graduate-level text is intended to provide an introduction to the common ground of its two subjects in topics such as operators and Hardy spaces,

closed operators, shift-invariance and causality, stability and stabilization, spaces of persistent signals, and delay systems.