Nonlinearities in finite dimensions can be linearized by projecting them intoinfinite dimensions. Unfortunately, often the linear operator techniques thatone would then use simply fail since the operators cannot be diagonalized. Thiscurse is well known. It also occurs for finite-dimensional linear operators. Wecircumvent it by developing a meromorphic functional calculus that candecompose arbitrary functions of nondiagonalizable linear operators in terms oftheir eigenvalues and projection operators. It extends the spectral theorem ofnormal operators to a much wider class, including circumstances in which polesand zeros of the function coincide with the operator spectrum. By allowing thedirect manipulation of individual eigenspaces of nonnormal andnondiagonalizable operators, the new theory avoids spurious divergences. Assuch, it yields novel insights and closed-form expressions across several areasof physics in which nondiagonalizable dynamics are relevant, includingmemoryful stochastic processes, open non unitary quantum systems, andfar-from-equilibrium thermodynamics. The technical contributions include the first full treatment of arbitrarypowers of an operator. In particular, we show that the Drazin inverse,previously only defined axiomatically, can be derived as the negative-one powerof singular operators within the meromorphic functional calculus and we give ageneral method to construct it. We provide new formulae for constructingprojection operators and delineate the relations between projection operators,eigenvectors, and generalized eigenvectors. By way of illustrating its application, we explore several, rather distinctexamples.
展开▼
