proved that S([phi]([rho])) = S([rho]) ([for all][rho] [member of] S(H)) if and only if there exists a

unitary operator U such that [phi]([rho]) = U[rho][U.sup.[dagger]], where [phi]([rho]) is a bistochastic quantum operation, i.e., [phi]([rho]) = [[summation].sub.i][E.sub.i][rho][E.sup.[dagger].sub.i] ([for all][rho] [member of] S(H)), and {[E.sub.i]} is a set of matrices known as operation elements.

Then [U.sub.p] is a

unitary operator on [F.sup.2] and [U.sup.-1.sub.p] = [U.sub.-p] [10, Proposition 2.3].

The question of whether Austin Powder is an "operator" under the Mine Act came up in 15 dockets with 22 citations and orders that were consolidated at the Federal Mine Safety and Health Review Commission for the purpose of determining whether Austin Powder and its subsidiaries constitute a "

unitary operator" for the purposes of the Mine Act.

exists for every

unitary operator U [member of] B (H), then T is compact and l = 0 holds.

If we think of the measurement as dictated by Schroedinger's equation, then we should have a

unitary operator [U.sub.M] [member of] H [cross product] [H.sub.I] with the following property:

Furthermore, because any

unitary operator on d qubits can be factored into a product of two-level

unitary operators on d qubits (Deutsch et al., 1995); and because these two-level

unitary operators on d qubits can in turn be exactly (i.e.

Let J be a

unitary operator in [L.sup.2]([??]) defined by

This is based on the Hadamard transformation, a

unitary operator on [l.sub.2] ([SIGMA]) whose matrix with respect to the standard basis is

Quantum stochastic calculus was designed to describe the dynamics of quantum processes and we propose that we use it to study the non commutative Merton-Black-Scholes model in the following formulation (notice that our model includes also the Poisson process): We replace (see [1] for details on quantization) the stock process {[X.sub.T] / t [greater than or equal to] 0} of the classical Black-Scholes theory by the quantum mechanical process [j.sub.t](X) = [U.sup.*.sub.t] X [cross product] 1 [U.sub.t] where, for each t [greater than or equal to] 0, [U.sub.t] is a

unitary operator defined on the tensor product H [cross product] [GAMMA]([L.sup.2]([R.sub.+], C)) of a system Hilbert space H and the noise Boson Fock space [GAMMA] = [GAMMA]([L.sup.2]([R.sub.+], C)) satisfying

Hence, A is a

unitary operator, with D(A) = [L.sup.2.sub.2[pi]].

For each n [greater than or equal to] 0, there exists a

unitary operator [U.sub.n] on [l.sup.2] (Z, H) such that

The question was posed and investigated in [7] and remains unanswered even if Banach-space power bounded operators are restricted to Hilbert-space contractions, where the problem is equivalently stated for

unitary operators (see Remark 3 below), and also if weak supercyclicity is strengthened to weak l-sequential supercyclicity: does there exist a weakly unstable and weakly l-sequentially supercyclic

unitary operator?