  Matrix Product State (MPS) wavefunctions have many applications in quantum
information and condensed matter physics. One application is to represent
states in the thermodynamic limit directly, using a small set of position
independent matrices. For this infinite MPS ansatz to be useful it is necessary
to be able to calculate expectation values, and we show here that a large class
of observables, including operators transforming under lattice translations as
eigenstates of arbitrary momentum $k$, can be represented in the Schur form of
a lower or upper triangular matrix and we present an algorithm for evaluating
such expectation values in the asymptotic limit. The sum or the product of two
such Schur operators is also a Schur operator, and is easily constructed to
give a simple method of constructing arbitrary polynomial combinations of
operators. Some simple examples are the variance $\langle (H-E)^2\rangle$ of an
infinite MPS, which gives a simple method of evaluating the accuracy of a
numerical approximation to a eigenstate, or a vertex operator $\langle
c^\dagger_{k_1} c^\dagger_{k_2} c_{k_4} c_{k_3}\rangle$. This approach is a
step towards improved algorithms for the calculation of dynamical properties
and excited states.
