for $ \phi {} \in {}{ L}^{ p}( \R {}^{ 2}, \C {}^{ 2}).$ For
${ p} = \infty {},$ we have continuity for the weak*-topology $ \sigma {}{ (}{ L}^ \infty {}( \R {}^{ 2 }, \C {}^{ 2}),{ L}^{ 1}( \R {}^{ 2}, \C {}^{ 2}){ )}.$
Each of the component functions $ \phi {}_{ 1}$ and $ \phi {}_{ 2}$
is transformed according to the action of a translation in $ \R {}^{ 2}.$\par
It follows from the Trotter product formula [K ] that the operator $- \alpha {} \partial _{ x} - { i}.{ m} \beta {}$
is also associated with with a uniformly bounded group of operators on
${ L}^ \infty {}( \R {}^{ 2}, \C {}^{ 2}),$ so Theorem XYZ shows that there are countably additive operator valued measures associated with equation (XY) and solutions to equation (XX) can be represented by integrals with respect to these measures.\par
are real-valued functions. The function $ \psi {}$ has values in ${\bf C}^{ N}.$
It is assumed that ${ P}_{ l} ,$ 1 $ \leq { l} \leq { d }$ have only real eigenvalues, and that they are simultaneously diagonalizable. The path space measures associated with the first order hyperbolic system (XXA) were first considered by T. Ichinose [Ich ] who examined properties of the fundamental solution of the system (XXA). \par
The alternate viewpoint using the Trotter product formula outlined above was formulated in [Jeff ] as an application of Theorem ZZZ. The essence of this approach is that there is a collection of dynamical systems, represented by translations along the
${ x}_{ i}-axes,$ 1 $ \leq { i} \leq { d}$ for the equations (XXA), that act independently on components of the state vector prior to suffering a
${ mixing}$ of components via a semigroup of operators , for example, the semigroup generated by the constant matrix
${ iQ}$ in equation (XXA). \par
The operator $\sum_{{ l =1}}^{{ d }} { }{ P}_{ l} \partial _{ x_{ i}{ }} $