Let [pi] : G [right arrow] B(H) be a projective, unitary representation of the group G into the

unitary operators on the Hilbert space H.

n] are unitary operators which yield the following unbounded linear functionals on [L.

n] is a unitary operator F such that the linear functionals [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] do not define bounded linear functionals.

Then the closure of T is a multiple of a unitary operator and the right-hand side of (2.

Motivated by the works of Mustari and Taylor [15,17], we have recently begun to study the mean ergodic theorem under the framework of RN modules [18] to obtain the mean ergodic theorem in the sense of convergence in probability, where we proved the mean ergodic theorem for a strongly continuous semigroup of random unitary operators defined on complete random inner product modules (briefly, complete RIP modules).

Let S be a complete random inner product module over C with base ([OMEGA], F,P), {U(t) : t [greater than or equal to] 0} a strongly continuous semigroup of random unitary operators on S and Po the random orthogonal projection onto the submodule So = {x [member of] S | U(t)x = x, [for all] t [greater than or equal to] 0}.

Zhang, Stone's representation theorem of a group of random unitary operators on complete complex random inner product modules, Sci.

Therefore, it is mandatory that the unitary operators (which are used for modelling quantum computations) are proved to have implementations that are based only on fault tolerant quantum gates.

But this set of gates it is not enough for universality as it doesn't spawn the whole set of unitary operators.

He explains vector spaces and bases, linear transformations and operators, eigenvalues, circles and ellipses, inner products, adjoints, Hermitian operators,

unitary operators, the wave equation, continuous spectre and the Dirac delta function, Fourier transforms, Green's and functions, and includes an appendix on matrix operations (new to this edition) and a full chapter on crucial applications.

In order to construct unitary operators that disperse the position of a particle, it is necessary to introduce an extra degree of freedom, known as chirality.

This is based on the Hadamard transformation, a unitary operator on [l.