Bessel functions and representation theory. I

In this paper a general theory of operator-valued Bessel functions is presented. These functions arise naturally in representation theory in the context of metaplectic representations, discrete series, and limits of discrete series for certain semi-simple Lie groups. In general, Bessel functions J_λ are associated to the action by automorphisms of a compact group U on a locally compact abelian group X, and are indexed by the irreducible representations λ of U that appear in the primary decomposition of the regular representation of U on L²(X). Then on the λ-primary constituent of L²(X), the Fourier transform is described by the Hankel transform corresponding to J_λ. More detailed information is available in the case in which (U, X) is an orthogonal transformation group which possesses a system of polar coordinates. In particular, when X=$\text{F}$^k×n,$\text{F}$ a real finite-dimensional division algebra, with k ? 2n and $\text{O}$(k, $\text{F}$), the representations λ of U are induced in a certain sense from representations π of GL(n, $\text{F}$). This leads to a characterization of J_λ as a reduced Bessel function defined on the component of 1 in GL(n, $\text{F}$) and to the connection between metaplectic representations and holomorphic discrete series for the group of biholomorphic automorphisms of the Siegel upper half-plane in the complexification of $\text{F}$^{n × n}.     

On some properties of convolution operators in K′1 and S′

Let $\text{H}$′ be either the space $\text{K}$′₁ of distributions of exponential growth or the space $\text{S}$′ of tempered distributions, and let $\text{O}$′_C($\text{H}$′:$\text{H}$′) be the space of convolution operators in $\text{H}$′. In each case $\text{H}$′ is the dual of a space $\text{H}$ of C^∞-functions which are in $\text{O}$′_C($\text{H}$′:$\text{H}$′). We establish necessary and sufficient conditions on the Fourier transform $\text{S}\text{?}$ of ? of S ? $\text{O}$′_C($\text{H}$′:$\text{H}$′) in order that every distribution u? $\text{O}$′_C($\text{H}$′:$\text{H}$′) with S1u?$\text{H}$ be in $\text{H}$. If $\text{H}$′ = $\text{K}$′₁, the condition is equivalent to S×H′¹=H′¹.     

Spectral theory of commuting self-adjoint partial differential operators

Let Ω denote a connected and open subset of $\text{R}$ⁿ. The existence of n commuting self-adjoint operators H₁,…, H_n on $\text{L}{}^2\text{(Ω)}$ such that each H_j is an extension of $\text{−}\text{i∂}\text{∂x}{}_{\mathrm{j}}$ (acting on $\text{C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(Ω))}$ is shown to be equivalent to the existence of a measure μ on $\text{R}$ⁿ such that f → $\text{}\text{̂}$tf (the Fourier transform of f) is unitary from $\text{L}{}^2\text{(Ω)}$ onto Ω. It is shown that the support of μ can be chosen as a subgroup of $\text{R}$ⁿ iff H₁,…, H_n can be chosen such that the unitary groups generated by H₁,…, H_n act multiplicatively on $\text{L}{}^2\text{(Ω)}$. This happens iff Ω (after correction by a null set) forms a system of representatives for the quotient of $\text{R}$ⁿ by some subgroup, i.e., iff Ω is essentially a fundamental domain.     

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 → ∞.     

On ranges of Lyapunov transformations

A Lyapunov transformation is a linear transformation on the set $\text{H}$_n of hermitian matrices H ? $\text{C}$^n,n of the form $\text{L}$_A(H) = A¹H + HA, where A ?$\text{C}$^n,n. Given a positive stable A ?$\text{C}$^n,n, the Stein-Pfeffer Theorem characterizes those K ? $\text{H}$_n for which K = $\text{L}$_B(H), where B is similar to A and H is positive definite. We give a new proof of this result, and extend it in several directions. The proofs involve the idea of a controllability subspace, employed previously in this context by Snyders and Zakai.     

Green's and Dirichlet spaces associated with fine Markov processes

This is the second paper in a series devoted to Green's and Dirichlet spaces. In the first paper, we have investigated Green's space $\text{K}$ and the Dirichlet space $\text{H}$ associated with a symmetric Markov transition function p_t(x, B). Now we assume that p is a transition function of a fine Markov process X and we prove that: (a) the space $\text{H}$ can be built from functions which are right continuous along almost all paths; (b) the positive cone $\text{K}$⁺ in $\text{K}$ can be identified with a cone M of measures on the state space; (c) the positive cone $\text{H}$⁺ in $\text{H}$ can be interpreted as the cone of Green's potentials of measures μ?M. To every measurable set B in the state space E there correspond a subspace $\text{K}$(B) of $\text{K}$ and a subspace $\text{H}$(B) of $\text{H}$. The orthogonal projections of $\text{K}$ onto $\text{K}$ and of $\text{H}$ onto $\text{H}$(B) can be expressed in terms of the hitting probabilities of B by the Markov process X. As the main tool, we use additive functionals of X corresponding to measures μ?M.     

Relative locality of derivations

Let H and K be symmetric linear operators on a C¹-algebra $\text{U}$ with domains D(H) and D(K). H is defined to be strongly K-local if $\text{ω(K(A)}{}^1\text{K(A)) = 0}$ implies $\text{ω(H(A)}{}^1\text{H(A)) = 0}$ for A?D(H) ∩ D(K) and ω in the state space of $\text{U}$, and H is completely strongly K-local if $\text{Ω(K(A)}{}^1\text{K(A))=0}$ implies $\text{Ω(H(A)}{}^1\text{H(A))=0}$ for A ∈ D(H) ∩ D(K) and Ω in the state of $\text{U}$, and H is cpmpletely strongly K-local if $\text{H??}{}_{\mathrm{n}}$ is $\text{K??}{}_{\mathrm{n}}$-local on U?M_n for all n ? 1, where $\text{1}$_n is the identity on the n × n matrices M_n. If $\text{U}$ is abelian then strong locality and complete strong locality are equivalent. The main result states that if τ is a strongly continuous one-parameter group of ¹-automorphisms of $\text{U}$ with generator δ₀ and δ is a derivation which commutes with τ and is completely strongly δ₀-local then δ generates a group α of ¹-automorphisms of $\text{U}$. Various characterizations of α are given and the particular case of periodic τ is discussed.     

Brauer groups and Amitsur cohomology for general commutative ring extensions

Exact couples are interconnected families of long exact sequences extending the short exact sequences usually derived from spectral sequences. This is exploited to give a long exact sequence connecting Amitsur cohomology groups $\text{H>}{}^{\mathrm{n}}\text{(}\text{S}\text{R}\text{, U)}$ (where U means the multiplicative group) and $\text{H}{}^{\mathrm{n}}\text{(}\text{S}\text{R}\text{, Pic)}$ and a third sequence of groups Hⁿ(J), for every faithfully flat commutative R-algebra S. This same sequence is derived in another way without assuming faithful flatness and Hⁿ(J) is identified explicitly as a certain subquotient of a group of isomorphism classes of pairs (P, α) with P a rank one, projective Sⁿ-module and α an isomorphism from the coboundary of P (inPicS^{n + 1}) toS^{n + 1}. (Here Sⁿ denotes repeated tensor product of S over R.) This last formulation allows us to construct a homomorphism of the relative Brauer group $\text{B(}\text{S}\text{R}\text{)}$ to H²(J) which is a monomorphism when S is faithfully flat over R, and an isomorphism when some S-module is faithfully projective over R. The first approach also identifies H²(J) with Ker[H²(R, U)→H²(S, U)], where H²(R, U) denotes the ordinary, Grothendieck cohomology (in the étale topology, for example).     

Inside the D-stable matrices

Let $\text{DS}{}_{\mathrm{n}}\text{(}\text{F}\text{)}$ denote the set of n×nD-stable matrices with entries from $\text{F}\text{?}\text{C}$. A characterization of the interior of $\text{DS}{}_{\mathrm{n}}\text{(}\text{F}\text{)}$ considered as subset of the topological space $\text{F}$ⁿ², is given for the cases $\text{F}\text{=}\text{R}\text{and}\text{C}$.     

On support properties of Lp-functions and their Fourier transforms

We give a criterion for the intersection of two projections in Hilbert space to be a projection of finite-dimensional range. This criterion is applied to Schrödinger operators in L²($\text{R}$ⁿ) and to the problem of determining whether there are functions f in L^v($\text{R}$ⁿ) such that both f and its Fourier transform have prescribed support.     

Complex dynamical variables for multiparticle systems with analytic interactions. I

The n-body problem is formulated as a problem of functional analysis on a Hilbert space $\text{G}$ whose elements are analytic functions of complex dynamical variables. It is assumed that the two-body interaction is local and spherically symmetric, and belongs to the two-particle space $\text{G}$. The n-body resolvent R(λ) is constructed with the help of Fredholm methods. The operator R(λ) on $\text{G}$ is associated with a family of operators R(λ, ?) on $\text{L}$² which are resolvents of closed linear operators H(?), the case ? = 0 corresponding to standard quantum mechanics. The spectrum of H(?) contains a set of parallel half-lines starting at the thresholds of scattering channels and making an angle 2? with the positive real axis. The half-lines are branch cuts of R(λ, ?), but matrix elements of R(λ, ?) can be continued analytically across these. The operator R(λ, ?) may have isolated poles. The location of these does not depend on ?. Each pole is associated with one or more eigenvectors of H(?) belonging to spaces $\text{G}$. There may be poles off the real axis, the location of a pole determining for which values of ? it is on the physical sheet of H(?). It is shown how poles off the real axis give rise to resonances in the scattering cross section, the shape of a resonance being as one would expect on the basis of a model in which the scattering takes place via a decaying compound state having an eigenvector of H(?) with complex energy as its wave function.     

Tempered distributions on Heisenberg groups whose convolution with Schwartz class functions is Schwartz class

Let N be a nilpotent Lie group and Q a tempered distribution on N. We say that Q is a left $\text{J}$-multiplier if convolution on the left by Q takes Schwartz class functions to Schwartz class functions; there is a similar definition for right $\text{J}$-multipliers. We show that if ? is an irreducible unitary representation of N, then one can define ρ(Q):$\text{H}$^∞:(ρ)→$\text{H}$^∞:(ρ) whenever Q is a left $\text{J}$-multiplier. The main results of the paper characterize left $\text{J}$-multipliers Q on Heisenberg groups in terms of the transform operators ?(Q) and show how this characterization can be used to find fundamental solutions of some left invariant differential operators. There is also an example of a left $\text{J}$-multiplier which is not a right $\text{J}$-multiplier.     

The optimization of the domain problem. I. Basic concepts

We introduce a representation of geometric shapes by vectors in an inner product space. A suitable renormalization procedure results in bounded regions of $\text{R}$ⁿ being represented by vectors of finite length.     

Operators as spectral integrals of operator-valued functions from the study of multivariate stationary stochastic processes

P. Masani and the author have previously answered the question, “When is an operator on a Hilbert space $\text{H}$ the integral of a complex-valued function with respect to a given spectral (projection-valued) measure?” In this paper answers are given to the question, “When is a linear operator from $\text{H}$^q to $\text{H}$^p the integral of a spectral measure?”; here the values of the integrand are linear operators from the square-summable q-tuples of complex numbers to the square-summable p-tuples of complex numbers, and our spectral measure for $\text{H}$^q is the “inflation” of a spectral measure for $\text{H}$. In the course of this paper, we make available tools for handling the spectral analysis of q-variate weakly stationary processes, 1 ≤ q ≤ ∞, which should enable researchers to deal in the future with the case q = ∞. We show as one application of our theory that if U = ∫(in0, 2π]e^?iθE(dθ) is a unitary operator on $\text{H}$ and if T is a bounded linear operator from $\text{H}$^q to $\text{H}$^q (1 ≤ q ≤ ∞) which is a prediction operator for each stationary process (Uⁿx)_?∞^∞ ?$\text{H}$^q (for each x = (xⁱ)_i^j ∈ $\text{H}$^q, Uⁿx = (Uⁿxⁱ)_i=1^q), then T is a spectral integral, ∫(_0,2π)]Φ(θ) E(dθ), and the Banach norm of T, |T|_B = ess sup |Φ(θ)|_B.     

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.     

Commuting self-adjoint partial differential operators and a group theoretic problem

In Rⁿ let Ω denote a Nikodym region (= a connected open set on which every distribution of finite Dirichlet integral is itself in $\text{L}{}^2\text{(Ω))}$. The existence of n commuting self-adjoint operators $\text{H}{}_1\text{,…, H}{}_{\mathrm{n}}\text{in}\text{L}{}^2\text{(Ω)}$ such that each H_j is a restriction of $\text{?i β}\text{βx}{}_{\mathrm{j}}$ (acting in the distribution sense) is shown to be equivalent to the existence of a set Λ ?Rⁿ such that the restrictions to Ω of the functions exp i ∑ λ_jx_j form a total orthogonal family in $\text{L}{}^2\text{(Ω)}$. If it is required, in addition, that the unitary groups generated by H₁,…, H_n act multiplicatively on $\text{L}{}^2\text{(Ω)}$, then this is shown to correspond to the requirement that Λ can be chosen as a subgroup of the additive group Rⁿ. The measurable sets Ω ?Rⁿ (of finite Lebesgue measure) for which there exists a subgroup Λ ?Rⁿ as stated are precisely those measurable sets which (after a correction by a null set) form a system of representatives for the quotient of Rⁿ by some subgroup Γ (essentially the dual of Λ).     

Geometric Lipschitz spaces and applications to complex function theory and nilpotent groups

It is known that a function on $\text{R}$ⁿ which can be well approximated by polynomials, in the mean over Euclidean balls, is Lipschitz smooth in the usual sense. In this paper an analogous theorem is proved in which $\text{R}$ⁿ is replaced by a set X, the averages over balls are replaced by a family of sublinear operators satisfying certain axioms, and the polynomials are replaced by a class of functions having certain regularity properties with respect to the averaging operators. Applications are given to function theory on domains in $\text{C}$ⁿ, to nilpotent Lie groups, and to the classical Euclidean case. The first application provides a characterization of the duals of Hardy spaces on the ball in $\text{C}$ⁿ.     

On the spectral synthesis problem for hypersurfaces of RN

Let F¹(Rⁿ) denote the Fourier algebra on $\text{R}$ⁿ, and $\text{D}$($\text{R}$ⁿ) the space of test functions on $\text{R}$ⁿ. A closed subset E of $\text{R}$ⁿ is said to be of spectral synthesis if the only closed ideal J in F¹(Rⁿ) which has E as its hull

$\text{h(J)={x ? R}{}^{\mathrm{n}}\text{:f(x)=0}\text{for all}\text{f ? J}}$

is the ideal

$\text{k(E)={f?F}{}^1\text{(R}{}^{\mathrm{n}}\text{):f(E)=0}}$

. We consider sufficiently regular compact subsets of smooth submanifolds of $\text{R}$ⁿ with constant relative nullity. For such sets E we give an estimate of the degree of nilpotency of the algebra $\text{(k(E)∩D(R}{}^{\mathrm{n}}\text{))}{}^{\mathrm{?}}\text{j(E)}$, where j(E) denotes the smallest closed ideal in F¹(Rⁿ) with hull E. Especially in the case of hypersurfaces this estimate turns out to be exact. Moreover for this case we prove that k(E)∩D(Rⁿ) is dense in k(E). Together this solves the synthesis problem for such sets.     

Extensions of unbounded 1-derivations in UHF C1-algebras

We consider unbounded ¹-derivations δ in UHF-C¹-algebras A=(∪^∞_n=1A_n)^?) with dense domain. If ?_n:A→A_n denotes the conditional expectations onto the finite type I factors $\text{A}$_n, then we introduce a weak-commutativity condition for δ and the sequence (?_n). As a consequence of this condition on δ we establish the existence of an extension derivation δ′ which is the infinitesimal generator of a strongly continuous one-parameter group, α: $\text{R}$ → Aut($\text{A}$), of ¹-automorphisms, i.e., $\text{δ′(x) =}\text{(d}\text{dt)}\text{α}{}_{\mathrm{t}}\text{(x)¦}{}_{\mathrm{t\; =\; 0}}$ for x?D(δ′). Special properties of α (alias δ′) are considered. We show that AF-algebras are associated to proper restrictions δ of derivations δ′ of product type. We then turn to the extendability problem for quasifree derivations in the CAR-algebra. There, extensions δ′ are calculated which generate strongly continuous semigroups of ¹-homomorphisms. These semigroups do not extend to one-parameter groups unless the implementing symmetric operator in one-particle space is already self-adjoint.