Reconstruction of a factor from measures on Takesaki's unitary equivalence relation

Let $\text{Ol}$?L^∞(S, μ) be a maximal abelian subalgebra of the factor $\text{F}$ on separable Hilbert space with modular involution J. ($\text{Ol}$∪ J$\text{Ol}$J)″ is represented naturally as L^∞(S × S, λ). If Takesaki's unitary equivalence relation $\text{R}$ ? S × S is not λ-null, it is a measure groupoid. If it is conull, and ($\text{Ol}$∪ J$\text{Ol}$J)″ is maximal abelian, $\text{F}$ and $\text{Ol}$ are reconstructed by the σ-left regular representation procedure. Examples show that these hypotheses are not always satisfied. An application shows that the L^∞ spectrum of a properly infinite ergodic transformation is null with respect to the L² spectrum.     

Nonlinear boundary value problems and operators TT1

We consider nonlinear boundary value problems of the type $\text{L? + N? = 0}$ for the existence of solutions. It is assumed that L is a 2nth-order linear differential operator in the real Hilbert space S = L²[a, b] which admits a decomposition of the form $\text{L =}{}^{\mathrm{TT1}}$ where T is an nth-order linear differential operator and N is a nonlinear operator defined on a subspace of S. The decomposition of L induces a natural decomposition of the generalized inverse of L. Using the method of “alternative problems,” we split the boundary value problem into an equivalent system of two equations. The theory of monotone operators and the theory of nonlinear Hammerstein equations are then utilized to consider the solvability of the equivalent system.     

An integral operator connected with the Helmholtz equation

Let Lu be the integral operator defined by $\text{(L}{}_{\mathrm{k?}}\text{)(x, y) = ∝ s ∝ ?(x′, y′)(}\text{e}{}^{\mathrm{ik?}}\text{?}\text{) dx′ dy′, (x, y) ? S}$ where S is the interior of a smooth, closed Jordan curve in the plane, k is a complex number with Re k ? 0, Im k ? 0, and ?² = (x ?x′)² + (y ? y′)². We define $\text{q(x, y) = [}\text{dist}\text{((x, y), ?S)]}\text{1}\text{2}\text{, (x, y) ? S; L}{}_2\text{(q, S) = {? : ∝ s ∝ ¦ ?(x, y)¦}{}^2\text{q(x, y) dx dy < ∞}; W}{}_2{}^1\text{(q, S) = {? : ? ? L}{}_2\text{(q, S),}\text{??}\text{?x}\text{,}\text{?f}\text{?y}\text{? L}{}_2\text{(q, S)}}$, where in the definition of W₂¹(q, S) the derivatives are taken in the sense of distributions. We prove that L_k is a continuous 1-l mapping of L₂(q, S) onto W₂¹(q, S).     

Linear transformations on matrices: the invariance of generalized permutation matrices—III

Let F be a field, F1 be its multiplicative group, and $\text{H}$ = {H:H is a subgroup of F1 and there do not exist a, b?F1 such that Ha+b?H}. Let D_n be the dihedral group of degree n, H be a nontrivial group in $\text{H}$, and τ_n(H) = {α = (α₁, α₂,…, α_n):α_i?H}. For σ?D_n and α?τ_n(H), let P(σ, α) be the matrix whose (i,j) entry is α_iδ_iσ(j) (i.e., a generalized permutation matrix), and

$\text{P(D}{}_{\mathrm{n}}\text{, H) = {P(σ, α):σ?D}{}_{\mathrm{n}}\text{, α?τ}{}_{\mathrm{n}}\text{(H)}}$

. Let M_n(F) be the vector space of all n×n matrices over F and $\text{T}$P(D_n, H) = {T:T is a linear transformation on M_n (F) to itself and T(P(D_n, H)) = P(D_n, H)}. In this paper we classify all T in $\text{T}$P(D_n, H) and determine the structure of the group $\text{T}$P(D_n, H) (Theorems 1 to 4). An expository version of the main results is given in Sec. 1, and an example is given at the end of the paper.     

Sequence entropy and mixing

The relationship between sequence entropy and mixing is examined. Let T be an automorphism of a Lebesgue space X, $\text{L}$₀ denote the set of all partitions of X possessing finite entropy, and $\text{S}$ denote the set of all increasing sequences of positive integers. It is shown that: (1) T is mixing /a2 sup_{A ? B}h_A(T, α) = H(α) for all B∈I and α∈Z₀. (2) T is weakly mixing /a2 sup_Ah_A(T, α) = H(α) for all α∈Z₀. (3) If T is partially mixing with constant $\text{c (1 ?}\text{1}\text{e}\text{< c < 1)}$, then sup_{A ? B}h_A(T, α) > cH(α) for all B∈I and nontrivial α∈Z₀. (4) If sup_{A ? B}h_A(T, α) > 0 for all B∈I and nontrivial α∈Z₀, then T is weakly mixing.     

Unbounded operators: Perturbations and commutativity problems

Let H be a complex Hilbert space, and let G_i, i = 1, 2, be closed and orthogonal subspaces of the product space H × H. The subspace G = G₁ ⊕ G₂ is called a (graph) perturbation. We give conditions for invariance of regular operators (R.O.) under graph perturbations: When is the perturbation of a R.O. again a R.O.? If N is a Hilbert space we consider R.O. (i.e., densely defined and closed operators T) in H=$\text{L}$(N) such that G(T)=G(S)⊕V$\text{H}$(M, where G denotes the graph, S is a decomposable operator in H, V a decomposable partial isometry such that the initial space of V(t) is equal to M a.e. t, and finally $\text{H}$(M) is the Hardy space of analytic $\text{L}$₂ vector functions with values in M ? N × N. Such operators T commute with the bilateral shift U; but, unless M = 0, T does not commute with $\text{U}{}^1$. Conversely, this is a canonical model for all R.O. with said commutativity properties. Moreover, the model is unique when T is given, and M = G(w) where w is a partial isometry in N. The detailed structure of the model is analyzed in the special case where dim N = dim M = 1. We relate the problem to a condition of Szeg? by showing that T is a $\text{R.O. iff}\text{∝}{}_0{}^{\mathrm{2\pi }}\text{log}\text{¦ V}{}_2\text{(t)¦ dt = ?∞,}\text{where}\text{V = (V}{}_1\text{, V}{}_2\text{)}$ is the partial isometry in the special case of dimension one. Szeg?'s conditions enters in a different way in the analysis of the case M = N × N, as well as in the spectral analysis of T. Our results provide an answer to a commutativity problem posed by Fuglede. If T is an arbitrary densely defined operator, and A?B(H) is normal, we prove two theorems stating conditions for the implication $\text{A}{}_{\mathrm{\cup }}\text{T ? A}{}_{\mathrm{\cup }}{}^1\text{T}$. These conditions cannot generally be relaxed.     

When is an operator the integral of a given spectral measure?

For a closed densely defined operator T on a complex Hilbert space $\text{H}$ and a spectral measure E for $\text{H}$ of countable multiplicity q defined on a σ-algebra $\text{B}$ over an arbitrary space Λ we give three conceptually differing but equivalent answers to the question asked in the title of the paper (Theorem 1.5). We then study the simplifications which accrue when T is continuous or when q = 1 (Sect. 4). With the aid of these results we obtain necessary and sufficient conditions for T to be the integral of the spectral measure of a given group of unitary operators parametrized over a locally compact abelian group Γ (Sect. 5). Applying this result to the Hilbert space $\text{H}$ of functions which are L₂ with respect to Haar measure for Γ, we derive a generalization of Bochner's theorem on multiplication operators (Sect. 6). Some results on the multiplicity of indicator spectral measures over Γ are also obtained. When Γ = $\text{R}$ we easily deduce the classical theorem about the commutant of the associated self-adjoint operator (Sect. 7).     

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.     

On an inequality of Tosio Kato for degenerate-elliptic operators

Let Ω be a domain in Rⁿ and T = ∑_{j,k = 1}ⁿ(?_j ? ib_j(x)) a_jk(x)(?_k ? ib_k(x)), where the a_jk and the b_j are real valued functions in $\text{C}{}^1\text{(Ω)}$, and the matrix (a_jk(x)) is symmetric and positive definite for every $\text{x ? Ω}$. If T₀ is the same as T but with b_j = 0, j = 1,…, n, and if u and Tu are in $\text{L}{}_{\mathrm{<mtext>loc</mtext>}}{}^1\text{(Ω)}$, then T. Kato has established the distributional inequality $\text{T}{}_0\text{¦ u ¦ ?}\text{Re[(sign}$ ū) Tu]. He then used this result to obtain selfadjointness results for perturbed operators of the form T ? q on Rⁿ. In this paper we shall obtain Kato's inequality for degenerate-elliptic operators with real coefficients. We then use this to get selfadjointness results for second order degenerate-elliptic operators on Rⁿ.     

A Euler-Poincaré characteristic for the open set lattices of finite topologies

Let $\text{T}$ be a finite topology. If P and Q are open sets of $\text{T}$ (Q may be the null set) then P is a minimal cover of Q provided Q ? P and there does not exist any open set R of $\text{T}$ such that Q ? R ? P. A subcollection $\text{D}$ of the open sets of $\text{T}$ is termed an i-discrete collection of $\text{T}$ provided $\text{D}$ contains every open O ∈ $\text{T}$ with the property that ? $\text{D}$ ? O ? ? $\text{D}$, $\text{D}$ contains exactly i minimal covers of ? $\text{D}$, and provided ?$\text{D}$ = ?{O | O ∈ $\text{D}$ and O is a minimal cover of ? $\text{D}$}. A single open set is a O-discrete collection. The number of distinct i-discrete collections of $\text{T}$ is denoted by p($\text{T}$, i). If there does not exist any i-discrete collection then p($\text{T}$,i) = 0, and this happens trivially for the case when i is greater than the number of points on which $\text{T}$ is defined. The object of this article is to establish the theorem: For any finite topology $\text{T}$, the quantity E($\text{T}$) = Σ_{i = 0}^∞ (?1)ⁱp($\text{T}$, i) = 1.     

The covering problem for the fields Q(−1)12 and Q(−3)12

Let k be a positive square free integer, $\text{N(?k)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ the ring of algebraic integers in $\text{Q(?k)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and S the unit sphere in C_n, complex n-space. If A₁,…, A_n are n linearly independent points of C_n then L = {u₁Au + … + u_nA_n} with $\text{u}{}_{\mathrm{r}}\text{∈ N(?k)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is called a k-lattice. The determinant of L is denoted by d(L). If L is a covering lattice for S, then $\text{θ(S, L) =}\text{V(S)}\text{d(L)}$ is the covering density. L is called locally (absolutely) extreme if θ(S, L) is a local (absolute) minimum. In this paper we determine unique classes of extreme lattices for k = 1 and k = 3.     

Pointwise convergence of eigenfunction expansions associated with ordinary differential operators

With an ordinary differential expression L = ∑ⁿ_k=0P_kD^k on an open interval I?$\text{r}$ is associated a selfadjoint operator H in a Hilbert space, possibly beyond $\text{K}$=L²(l). The set $\text{D}$H∩$\text{K}$ only depends on the generalized spectral family associated with H. It is shown that the (differentiated) eigenfunction expansion given by H converges uniformly on compact subintervals of l for functions in $\text{D}$(H)∩$\text{L}$ In case H is a semibounded selfadjoint operator in $\text{K}$=L²T, a similar result is proved for functions in $\text{D}$|H|, which is the set of all $\text{K}$∈$\text{K}$ for which there exists a sequence f_n∈(H) such that f_n→f in $\text{H}$ and (H(f_n ? f_m), f_n ? f_m → 0 as n, m → ∞.     

Permutation-operator inequalities on certain symmetry classes of tensors

If A∈T(m, N), the real-valued N-linear functions on E_m, and σ∈S_N, the symmetric group on {…,N}, then we define the permutation operator P_σ: T(m, N) → T(m, N) such that P_σ(A)(x₁,x₂,…,x_N = A(x_σ(1),x_σ(2),…, x_σ(N)). Suppose Σ^q_i=1n_i = N, where the n_i are positive integers. In this paper we present a condition on σ that is sufficient to guarantee that 〈P_σ(A₁?A₂???A_q),A₁?A₂?? ? A_q〉 ? 0 for A_i∈S(m, n_i), where S(m, n_i) denotes the subspace of T(m, n_i) consisting of all the fully symmetric members of T(m, n_i). Also we present a broad generalization of the Neuberger identity which is sometimes useful in answering questions of the type described below. Suppose G and H are subgroups of S_N. We let T_G(m, N) denote all A∈T(m, N) such that P_σ(A) = A for all σ∈G. We define the symmetrizer $\text{S}$_G: T(m, N)→T_G(m,N) such that $\text{S}{}_{\mathrm{G}}\text{(A) = 1/|G|Σ}{}_{\mathrm{\sigma \in G}}\text{P}{}_{\mathrm{\sigma }}\text{(A)}$. Suppose H is a subgroup of G and A∈T_H(m, N). Clearly $\text{6}\text{S}{}_{\mathrm{<mtext>G</mtext>}}\text{6(A) 6? 6A6.}$ We are interested in the reverse type of comparison. In particular, if D is a suitably chosen subset of T_H(m,N), then can we explicitly present a constant C>0 such that $\text{6}\text{S}{}_{\mathrm{G}}\text{(A)6?C6A6}$ for all A∈D?     

On constant discrete programming problems

Let H be a subset of the set S_n of all permutations

$\text{1}\text{2}\text{???}\text{n}\text{s(1)}\text{s(2)}\text{???}\text{s(n)}$

C=6c_ij6 a real n?n matrix L_c(s)=c_1s(1)+c_2s(2)+???+c_ns(n) for s ? H. A pair (H, C) is the existencee of reals a₁,b₁,a₂,b₂,…a_n,b_n, for which c_ij=a₁+b_j if (i,j)?D(H), where D(H)={(i,j):(?h?H)(j=h(i))}.For a pair (H,C) the specifity of it is proved in the case, when H is either a special cyclic class of permutations or a special union of cyclic classes. Specific pairs with minimal sets H are in some sense described.     

Existence and boundedness of solutions of abstract nonlinear integrodifferential equations of nonconvolution type

Existence and boundedness theorems are given for solutions of nonlinear integrodifferential equations of type $\text{d}\text{dt}\text{u(t) + Bu(t) + ∝}{}_0{}^{\mathrm{t}}\text{a(t, s) Au(s) ds ? f(t) (t > 0)}$, (1.1) u(0) = u₀, Here A and B are nonlinear, possibly multivalued, operators on a Banach space W and a Hilbert space H, where W ? H. The function f (0, ∞) → H and the kernel a(t, s): $\text{R}$ × $\text{R}$ → $\text{R}$ are known functions. The results of this paper extend the results of Crandall, Londen, and Nohel [4] for equation (1.1). They assumed the kernel to be of the type a(t, s) = a(t ? s). We relax this assumption and obtain similar results. Examples of kernels satisfying the conditions we require are given in section 4.     

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.     

Projections on spaces of invariant functions

Let (X, ∑, μ) be a measure space and S be a semigroup of measure-preserving transformations T:X → X. In case μ(X) < ∞, Aribaud [1] proved the existence of a positive contractive projection P of L¹(μ) such that for every $\text{? ? L}{}^1\text{(μ), Pf}$ belongs to the closure $\text{C}{}_1\text{(?)}\text{in}\text{L}{}^1\text{(μ)}$ of the convex hull $\text{C(?)}$ of the set {$\text{? ○ T:T ? S}$}. In this paper we extend this result in three directions: we consider infinite measure spaces, vector-valued functions, and L^p spaces with 1 ? p < ∞, and prove that P is in fact the conditional expectation with respect to the σ-algebra Λ of sets of ∑ which are invariant with respect to all T?S.     

The strong solution of the Monge–Ampère equation on the Wiener space for log-concave densities

Let (W,H,μ) be an abstract Wiener space, assume that $\text{d}\text{ν=L}\text{d}\text{μ}$ is a second probability measures on $\text{(W,}\text{B}\text{(W))}$ such that $\text{L=}\text{1}\text{c}\text{exp}\text{?f}$, with $\text{f∈}\text{D}{}_{\mathrm{2,1}}$ lower bounded and H-convex. Let $\text{T=I}{}_{\mathrm{W}}\text{+??,}\text{?∈}\text{D}{}_{\mathrm{2,1}}$, be the solution of the Monge problem transporting μ to ν and realizing the H-Wasserstein distance between μ and ν. We prove that $\text{?∈}\text{D}{}_{\mathrm{2,2}}$ hence the Gaussian Jacobian $\text{Λ=}\text{det}{}_2\text{(I+?}{}^2\text{?)}\text{exp}\text{{}\text{L}\text{??1/2|??|}{}_{\mathrm{H}}{}^2\text{}}$ is well-defined and T is the strong solution of the Monge–Ampère equation ΛL°T=1 a.s. on W. To cite this article: D. Feyel, A.S. Üstünel, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

A limit theorem for perturbed operator semigroups with applications to random evolutions

Let U(t) and S(t) be strongly continuous contraction semigroups on a Banach space L with infinitesimal operators A and B, respectively. Suppose the closure of A + αB generates a semigroup T_α(t). The behavior of T_α(t) as α goes to infinity is examined. In particular, suppose S(t) converges strongly to P. If the closure of PA generates a semigroup T(t) on $\text{R}$(P), then T_α(t) goes to T(t) on $\text{R}$(P). If PA = 0 and if BVf = ?f for fε$\text{N}$(P), conditions are given that imply T_α(αt) converges on $\text{R}$(P) to a semigroup generated by the closure of PAVA.The results are used to obtain new and known limit theorems for random evolutions, which in turn give approximation theorems for diffusion processes.