Representation of functions of Markov processes as solutions of stochastic equations

Let X_t be a homogeneous Markov process generated by the weak infinitesimal operator A. Let $\text{H}$ be the class of functions f such that f, f²?D_A, the domain of A. The main result of this paper states that for ? ∈ $\text{H}$ can be represented by a stochastic integral and other terms. If the process is generated by a second order differential operator (with ‘poor’ coefficients possibly) on C₀²(R^d) then the process itself can be represented as the solution of an Itô stochastic differential equation.     

Construction of a unique self-adjoint generator for a symmetric local semigroup

A definition is given of a symmetric local semigroup of (unbounded) operators P(t) (0 ? t ? T for some T > 0) on a Hilbert space $\text{H}$, such that P(t) is eventually densely defined as t → 0. It is shown that there exists a unique (unbounded below) self-adjoint operator H on $\text{H}$ such that P(t) is a restriction of e^?tH. As an application it is proven that H₀ + V is essentially self-adjoint, where e^?tH₀ is an L^p-contractive semigroup and V is multiplication by a real measurable function such that V ∈ L^{2 + ε} and e^?δV ∈ L¹ for some ε, δ > 0.     

Abstract equipartition of energy theorems

Let H be a self-adjoint operator on a complex Hilbert space $\text{H}$. The solution of the abstract Schrödinger equation $\text{idu}\text{dt}\text{= Hu}$ is given by u(t) = exp(?itH)u(0). The energy E = ∥u(t)∥² is independent of t. When does the energy break up into different kinds of energy E = ∑_{j = 1}^NE_j(t) which become asymptotically equipartitioned ? (That is, $\text{E}{}_{\mathrm{j}}\text{(t) →}\text{E}\text{N}\text{as}\text{t → ± ∞}$ for all j and all data u(0).) The “classical” case is the abstract wave equation $\text{d}{}^2\text{v}\text{dt}{}^2\text{+ A}{}^2\text{v = 0}\text{with}\text{A}$ self-adjoint on $\text{H}$₁. This becomes a Schrödinger equation in a Hilbert space $\text{H}$ (essentially $\text{H}$ is two copies of $\text{H}$₁), and there are two kinds of associated energy, viz., kinetic and potential. Two kinds of results are obtained. (1) Equipartition of energy is related to the C¹-algebra approach to quantum field theory and statistical mechanics. (2) Let A₁,…, A_N be commuting self-adjoint operators with N = 2 or 4. Then the equation $\text{Π}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{N}}\text{(}\text{d}\text{dt}\text{? iA}{}_{\mathrm{j}}\text{) u(t) = 0}$ admits equipartition of energy if and only if exp(it(A_j ? A_k)) → 0 in the weak operator topology as t → ± ∞ for j ≠ k.     

Dissipators for symmetric quasi-free dynamical semigroups on the CAR algebra

Let L be a negative self-adjoint bounded operator on a Hilbert space $\text{H}$, and p a projection on $\text{H}$ with pLp trace class, and let {T_t: t ? 0} be the extension of {e^tL: t ? 0} to a strongly continuous semigroup of completely positive quasi-free unital maps of Fock type on the fermion algebra A$\text{H}$ built over $\text{H}$. Then it is shown that there exists a strongly continuous self-adjoint contraction semigroup {G_t: t ? 0} on the Hilbert space of the GNS decomposition of the quasi-free state gw_p such that in the representation of that state: T_t ? G_t(·)G_t, t ?0.     

Self-adjointness of relatively bounded quadratic forms and operators

We consider a Hilbert space $\text{H}$ on which is given a positive self-adjoint operator H. For densely defined bilinear forms or operators A we obtain conditions which ensure that A is an operator, that A is self-adjoint and that e^iAt leaves D(H^r) invariant with H^re^iAt strongly differentiable.     

Equivalent measures of dependence

The maximal correlation between a pair of σ-fields $\text{A}$ and $\text{B}$ becomes arbitrarily small as sup{|P(A ? B) ? P(A) P(B)|/[P(A) P(B)]^1/2, A ∈ $\text{A}$, B ∈ $\text{B}$, P(A) > 0, P(B) > 0} becomes sufficiently small.     

Unitarization of symplectics and stability for causal differential equations in Hilbert space

Let Sp($\text{H}$) be the symplectic group for a complex Hibert space $\text{H}$. Its Lie algebra sp($\text{H}$) contains an open invariant convex cone C₀; each element of C₀ commutes with a unique sympletic complex structure. The Cayley transform $\text{C}$: X∈ sp($\text{H}$)→(I + X)¹∈ Sp($\text{H}$) is analyzed and compared with the exponential mapping. As an application we consider equations of the form $\text{(}\text{d}\text{dt}\text{) S = A(t)S,}\text{where}\text{t → A(t) ?}\text{C}\text{?}{}_0$ is strongly continuous, and show that if ∝_?∞^∞ ∥A(t)∥ dt < 2 and ∝_{? t8}^∞A(t) dt?C₀, the (scattering) operator

, where S_t′(t) is the solution such that S_t′(t′) = I, is in the range of $\text{B}$ restricted to C₀. It follows that S leaves invariant a unique complex structure; in particular, it is conjugate in Sp($\text{H}$) to a unitary operator.     

Injectivity and operator spaces

Let B(H) be the bounded operators on a Hilbert space H. A linear subspace R ? B(H) is said to be an operator system if 1 ?R and R is self-adjoint. Consider the category $\text{b}$ of operator systems and completely positive linear maps. R ∈ $\text{C}$ is said to be injective if given A ? B, A, B ∈ $\text{C}$, each map A → R extends to B. Then each injective operator system is isomorphic to a conditionally complete C¹-algebra. Injective von Neumann algebras R are characterized by any one of the following: (1) a relative interpolation property, (2) a finite “projectivity” property, (3) letting M_m = B(C^m), each map R → N ? M_m has approximate factorizations R → M_n → N, (4) letting K be the orthogonal complement of an operator system N ? M_m, each map $\text{M}{}_{\mathrm{m}}\text{K}\text{→ R}$ has approximate factorizations $\text{M}{}_{\mathrm{m}}\text{K}\text{→ M}{}_{\mathrm{n}}\text{→ R}$. Analogous characterizations are found for certain classes of C¹-algebras.     

A characterization of the minimum cycle mean in a digraph

Let C = (V,E) be a digraph with n vertices. Let f be a function from E into the real numbers, associating with each edge e ∈ E a weight$\text{?(e)}$. Given any sequence of edges σ = e₁,e₂,…,e_p define w(σ), the weight of σ, as $\text{∑}\text{i =}\text{1}\text{p}\text{?(e}{}_{\mathrm{i}}\text{)}$, and define m(σ), the mean weight of σ, as w(σ)?p. Let $\text{λ}{}^1\text{=}\text{min}{}_{\mathrm{C}}\text{m(C)}$ where C ranges over all directed cycles in G; λ¹ is called the minimum cycle mean. We give a simple characterization of λ¹, as well as an algorithm for computing it efficiently.     

Essential unitarization of symplectics and applications to field quantization

Symplectic operators satisfying generic and group-invariant (spectral) positivity conditions are studied; the theory developed is applied and illustrated to determine the unique invariant frequency decomposition (equivalently, linear quantization with invariant vacuum state) of the Klein-Gordon equation in non-static spacetimes. Let (H, Ω) be any linear topological symplectic space such that there exists a real-linear and topological isomorphism of H with some complex Hilbert space carrying Ω into the imaginary part of the scalar product. Then any bounded invertible symplectic S ∈ Sp(H) (resp. bounded infinitesimally symplectic A ∈ sp(H)) which satisfies $\text{Ω(Sv, v) > 0}$ (resp. $\text{Ω(Av, v) > 0}$) for all nonzero v ω H, where S + I is invertible, is realized uniquely and constructively as a unitary (resp. skewadjoint) operator in a complex Hilbert space which depends in general on the operator and typically only densely intersects H. The essentially unique weakly and uniformly closed invariant convex cones in sp(H) are determined, extending previously known results in the finite-dimensional case. A notion of “skew-adjoint extension” of a closed semi-bounded infinitesimally symplectic operator is defined, strictly including the usual notion of positive self-adjoint extension in a complex Hilbert space; all such skew-adjoint extensions are parametrized, as in the von Neumann or Birman-Krein-Vishik theories. Finally, the unique complex Hilbertian structure—formulated on the space of solutions of the covariant Klein-Gordon equation in generic conformal perturbations of flat space—is uniquely determined by invariance under the scattering operator. The invariant Hilbert structure is explicitly calculated to first order for an infinite-dimensional class of purely time-dependent metric perturbations, and higher-order contributions are rigorously estimated.     

Translation-invariant linear forms on C0(G), C(G), Lp(G) for noncompact groups

Suppose G is a locally compact noncompact group. For abelian such G's, it is shown in this paper that L¹(G), C(G), and L^∞(G) always have discontinuous translation-invariant linear forms(TILF's) while C₀(G) and L^p(G) for 1 < p < ∞ have such forms if and only if $\text{G}\text{H}$ is a torsion group for some open σ-compact subgroup H of $\text{G}$. For σ-compact amenable G's, all the above spaces have discontinuous left TILF's.     

Spectral properties of zeroth-order pseudodifferential operators

Let Q be a self-adjoint, classical, zeroth order pseudodifferential operator on a compact manifold X with a fixed smooth measure dx. We use microlocal techniques to study the spectrum and spectral family, {E_S}_{S∈$\text{R}$} as a bounded operator on L²(X, dx).Using theorems of Weyl (Rend. Circ. Mat. Palermo, 27 (1909), 373–392) and Kato (“Perturbation Theory for Linear Operators,” Springer-Verlag, 1976) on spectra of perturbed operators we observe that the essential spectrum and the absolutely continuous spectrum of Q are determined by a finite number of terms in the symbol expansion. In particular Spec_ESSQ = range(q(x, ξ)) where q is the principal symbol of Q. Turning the attention to the spectral family {E_S}_{S∈$\text{R}$}, it is shown that if $\text{dE}\text{ds}$ is considered as a distribution on $\text{R}$×X×X it is in fact a Lagrangian distribution near the set $\text{{σ=0}?}\text{T}{}^1\text{(}\text{R}\text{×X×X)}\text{0}$ where (s, x, y, σ, ξ,η) are coordinates on T¹($\text{R}$×X×X) induced by the coordinates (s, x, y) on $\text{R}$×X×X. This leads to an easy proof that$\text{?(Q)}$ is a pseudodifferential operator if ?∈C^∞($\text{R}$) and to some results on the microlocal character of E_s. Finally, a look at the wavefront set of $\text{dE}\text{ds}$ leads to a conjecture about the existence of absolutely continuous spectrum in terms of a condition on q(x, ξ).     

Nonlinear eigenvalue problems with positively convex operators

We consider the equation u = λAu (λ > 0), where A is a forced isotone positively convex operator in a partially ordered normed space with a complete positive cone K. Let Λ be the set of positive λ for which the equation has a solution u?K, and let Λ₀ be the set of positive λ for which a positive solution—necessarily the minimum one—can be obtained by an iteration u_n = λAu_n?1, u₀ = 0. We show that if K is normal, and if Λ is nonempty, then Λ₀ is nonempty, and each set Λ₀, Λ is an interval with $\text{inf}\text{(Λ}{}_0\text{) =}\text{inf}\text{(Λ) = 0}\text{and sup}\text{(Λ}{}_0\text{) =}\text{sup}\text{(Λ) (= λ1,}\text{say}\text{)}$; but we may have λ1 ? Λ₀ and λ1 ? Λ. Furthermore, if A is bounded on the intersection of K with a neighborhood of 0, then Λ₀ is nonempty. Let u⁰(λ) = lim_n→∞(λA)ⁿ(0) be the minimum positive fixed point corresponding to λ ? Λ₀. Then u⁰(λ) is a continuous isotone convex function of λ on Λ₀.     

Hyperspaces of compact sets in metric linear spaces

We write 2^x for the hyperspace of all non-empty compact sets in a complete metric linear space X topologized by the Hausdorff metric. Using the notation $\text{F}$(X) = {A ϵ 2^X: A is finite}, l^f₂ = {x} = (x_i) ϵ l₂: x_i = 0 for almost all i}, and l^σ₂ = {x = (x _i) ϵ l₂:σ^∞_i=1 (ix_i)² < ∞}, we have the following theorem:A family $\text{G}$ ⊂$\text{F}$(X) is homeomorphic to l^f₂ if $\text{G}$ is σ-fd-compact, the closure $\text{G}$ of $\text{G}$ in 2^x is not locally compact and if whenever A, B ∈ $\text{G}$, λ ∈ [0, 1] and C ⊂ λA + (1 - λ)B with card C⩽ max{card A, card B} then C ϵ $\text{G}$.Moreover, for any G_δ-AR-set $\text{G}$$\text{G}$ of $\text{G}$ with $\text{G}$$\text{G}$ ⊃ $\text{G}$ we have ($\text{G}$$\text{G}$, $\text{G}$)≅(l₂, l^ƒ₂).Similar conditions for hyperspaces to be homeomorphic to l^σ₂ are also established.     

Nonlinear Volterra equations on a Hilbert space

We consider equations of the form, u(t) = ? ∝₀^tA(t ? τ)g(u(τ)) dτ + ?(t) (I) on a Hubert space $\text{H}$. A(t) is a family of bounded, linear operators on $\text{H}$ while g is a transformation on $\text{D}$_g ?$\text{H}$ which can be nonlinear and unbounded. We give conditions on A and g which yield stability and asymptotic stability of solutions of (I). It is shown, in particular, that linear combinations with positive coefficients of the operators e^Mt and ?e^Mtsin Mt where M is a bounded, negative self-adjoint operator on $\text{H}$ satisfy these conditions. This is shown to yield stability results for differential equations of the form, $\text{Q (}\text{d}\text{dt}\text{) u = ? P (}\text{d}\text{dt}\text{) g(u(t)) + χ(t)}$, on $\text{H}$.     

Singular perturbations in the interaction representation

A general method for treating highly singular perturbations V of self-adjoint operators H in Hilbert space is applied to the case of perturbations of $\text{(?}\text{id}\text{dx}\text{)}{}^{\mathrm{n}}$ in L₂ ($\text{R}$¹ by (multiplications by) distributions. A self-adjoint operator H_V that agrees with H + V in the usual sense when V is sufficiently regular, and is moreover a continuous function of V, within the class of distributions under consideration, in the strong operator topology for unbounded self-adjoint operators, is shown to exist. This operator H_V need not be semi-bounded, or determined by a sesquilinear form associated with H + V. The method proceeds by construction of the corresponding unitary propagator in the interaction representation, essentially e^?itHVe^itH, which is shown to be expressible as a uniformly convergent perturbative series for small times.     

The solutions of a variational inequality as critical points of a real valued smooth function

Let H be a real Hilbert space, A : H → H a self-adjoint continuous linear operator. Given a variational inequality related to A, we construct a Lipschitz continuously differentiable function F:H→$\text{R}$ so that the critical points of F are the solutions of the variational inequality.     

A surprising property of the least eigenvalue of a graph

Let λ(G) be the least eigenvalue of a graph G. A real number r has the induced subgraph property provided λ(G)<r implies G has an induced subgraph H with λ(H)=r. It is shown that the only numbers with the induced subgraph property are 0, ?1, ?$\text{2}$, and ?2.     

Ein operatorwertiger Hahn-Banach Satz

We generalize Arveson's extension theorem for completely positive mappings [1] to a Hahn-Banach principle for matricial sublinear functionals with values in an injective C¹-algebra or an ideal in B($\text{H}$). We characterize injective W¹-algebras by a matricial order condition. We illustrate the matricial Hahn-Banach principle by three applications: (1) Let $\text{A, B, b}$ be unital C¹-algebras, $\text{b}$ a subalgebra of $\text{A}$ and $\text{B, B}$ injective. If ?: $\text{A}$ → $\text{B}$ is a completely bounded self-adjoint $\text{b}$-bihomomorphism, then it can be expressed as the difference of two completely positive $\text{b}$-bihomomorphism. (2) Let $\text{M}$ be a W¹-algebra, containing 1_$\text{H}$, on a Hilbert space $\text{H}$. If $\text{M}$ is finite and hyperfinite, there exists an invariant expectation mapping P of B($\text{H}$) onto $\text{M}$′. P is an extension of the center trace. (3) Combes [7] proved, that a lower semicontinuous scalar weight on a C¹-algebra is the upper envelope of bounded positive functionals. We generalize this result to unbounded completely positive mappings with values in an injective W¹-algebra.     

Approximation of isolated eigenvalues of general singular ordinary differential operators

Let A be a self-adjoint operator defined by a general singular ordinary differential expression τ on an interval (a, b), ? ∞ ≤ a < b ≤ ∞. We show that isolated eigenvalues in any gap of the essential spectrum of A are exactly the limits of eigenvalues of suitably chosen self-adjoint realizations A_n of τ on subintervals (a_n, b_n) of (a, b) with a_n → a, b_n → b. This means that eigenvalues of singular ordinary differential operators can be approximated by eigenvalues of regular operators. In the course of the proof we extend a result, which is well known for quasiregular differential expressions, to the general case: If the spectrum of A is not the whole real line, then the boundary conditions needed to define A can be given using solutions of (τ ? λ)u = 0, where λ is contained in the regularity domain of the minimal operator corresponding to τ.