Conformal field theory
The Hilbert space of physical states is a unitary module of the Virasoro algebra corresponding to a fixed value of "c". Stability requires that the energy spectrum of the Hamiltonian be nonnegative. The modules of interest are the highest weight modules of the Virasoro algebra.
Spectrum (functional analysis)
For example, consider the Hilbert space formula_24, that consists of all bi-infinite sequences of real numbers
that have a finite sum of squares formula_26. The bilateral shift operator formula_2 simply displaces every element of the sequence by one position; namely if formula_28 then formula_29 for every integer formula_30. The eigenvalue equation formula_31 has no solution in this space, since it implies that all the values formula_32 have the same absolute value (if formula_33) or are a geometric progression (if formula_34); either way, the sum of their squares would not be finite. However, the operator formula_20 is not invertible if formula_36. For example, the sequence formula_37 such that formula_38 is in formula_24; but there is no sequence formula_40 in formula_24 such that formula_42 (that is, formula_43 for all formula_30).