Beyond the spectral theorem: Spectrally decomposing arbitrary functions of nondiagonalizable operators

Abstract

Nonlinearities in finite dimensions can be linearized by projecting them into infinite dimensions. Unfortunately, often the linear operator techniques that one would then use simply fail since the operators cannot be diagonalized. This curse is well known. It also occurs for finite-dimensional linear operators. We circumvent it by developing a meromorphic functional calculus that can decompose arbitrary functions of nondiagonalizable linear operators in terms of their eigenvalues and projection operators. It extends the spectral theorem of normal operators to a much wider class, including circumstances in which poles and zeros of the function coincide with the operator spectrum. By allowing the direct manipulation of individual eigenspaces of nonnormal and nondiagonalizable operators, the new theory avoids spurious divergences. As such, it yields novel insights and closed-form expressions across several areas of physics in which nondiagonalizable dynamics are relevant, including memoryful stochastic processes, open nonunitary quantum systems, and far-from-equilibrium thermodynamics.The technical contributions include the first full treatment of arbitrary powers of an operator. In particular, we show that the Drazin inverse, previously only defined axiomatically, can be derived as the negative-one power of singular operators within the meromorphic functional calculus and we give a general method to construct it. We provide new formulae for constructing projection operators and delineate the relations between projection operators, eigenvectors, and generalized eigenvectors.By way of illustrating its application, we explore several, rather distinct examples. First, we analyze stochastic transition operators in discrete and continuous time. Second, we show that nondiagonalizability can be a robust feature of a stochastic process, induced even by simple counting. As a result, we directly derive distributions of the Poisson process and point out that nondiagonalizability is intrinsic to it and the broad class of hidden semi-Markov processes. Third, we show that the Drazin inverse arises naturally in stochastic thermodynamics and that applying the meromorphic functional calculus provides closed-form solutions for the dynamics of key thermodynamic observables. Fourth, we show that many memoryful processes have power spectra indistinguishable from white noise, despite being highly organized. Nevertheless, whenever the power spectrum is nontrivial, it is a direct signature of the spectrum and projection operators of the process' hidden linear dynamic, with nondiagonalizable subspaces yielding qualitatively distinct line profiles. Finally, we draw connections to the Ruelle-Frobenius-Perron and Koopman operators for chaotic dynamical systems. PACS numbers: 02.50.-r 05.45.Tp 02.50.Ey 02.50.Ga... the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.A. Einstein [1, p. 165]

Informational and Causal Architecture of Continuous-time Renewal Processes

“…Similar technical difficulties arise more generally when describing minimal maximally predictive models of other continuous-time, discrete-symbol processes that are not the continuous-time Markov processes analyzed in Ref. [6].…”

“…(6). Otherwise, arrows denote allowed transitions back to the reset state or "0 node" (solid black circle at left), denoting that an event occurred.…”

“…Some progress has been made on understanding minimal maximally predictive models of discrete-time, continuous-output processes; e.g., see Refs. [4][5][6]. Relatively less is understood about minimal maximally predictive models of continuous-time, discrete-output processes, beyond those with exponentially decaying state-dwell times [6].…”

“…[4][5][6]. Relatively less is understood about minimal maximally predictive models of continuous-time, discrete-output processes, beyond those with exponentially decaying state-dwell times [6]. The following is a first attempt at a remedy that complements the spectral methods developed in Ref.…”

“…The following is a first attempt at a remedy that complements the spectral methods developed in Ref. [6], as we address the less tractable case of uncountably infinite causal states. * semarzen@mit.edu † chaos@ucdavis.edu…”