Uncertainty principle for operators commuting with translations. II

Continuation of the authors' paper (RZhMat., 1980, 4B820). Convolution operators with semirational symbols (s.s.) are studied. Uniqueness theorems are proved for logarithmic potentials, as well as compatibility theorems for pairs of equations where K is a kernel with s.s., E is a sufficiently “sparse” subset of the line, f is an “unknown” function. Versions are considered of the “two constants theorem” of Hadamard, relating to uniqueness properties of operators with s.s.     

Continuity of linear operators commuting with shifts

It is shown that linear operators on a wide class of function and measure spaces on [0, ∞) which commute with the right shift operators are necessarily continuous. Indeed, commutativity with a single nonidentity shift is often sufficient, and the results are valid for more general cones. As a consequence characterizations of perfect operators and other multipliers are obtained, and applied to prove an extension of a result of Diamond for derivations on measure algebras.     

On operators commuting with Toeplitz operators modulo the compact operators

We prove that an operator on H² of the disc commutes modulo the compacts with all analytic Toeplitz operators if and only if it is a compact perturbation of a Toeplitz operator with symbol in H^∞ + C. Consequently, the essential commutant of the whole Toeplitz algebra is the algebra of Toeplitz operators with symbol in QC. The image in the Calkin algebra of the Toeplitz operators with symbol in H^∞ + C is a maximal abelian algebra. These results lead to a characterization of automorphisms of the algebra of compact perturbations of the analytic Toeplitz operators.     

A generalization to locally compact abelian groups of a spectral problem for commuting partial differential operators

Let G be a locally compact abelian group, and let Ω be an open relatively compact subset of positive Haar measure. Let Λ be a subset of the dual group G such that the restriction to Ω of Λ(·) for λ?Λ constitute an orthonormal basis for L₂(Ω) with normalized measure. We show that the pair Ω,Λ can be characterized completely in terms of group theory and the geometry of fundamental domains for discrete subgroups. Proofs are only sketched. The relationship to partial differential operators is pointed out.     

Mahler measure and entropy for commuting automorphisms of compact groups

We compute the joint entropy ofd commuting automorphisms of a compact metrizable group. LetR _d = [u ₁ ^±1 ,...,[d ₁ ^±1 ] be the ring of Laurent polynomials ind commuting variables, andM be anR _d -module. Then the dual groupX _M ofM is compact, and multiplication onM by each of thed variables corresponds to an action _M of ^d by automorphisms ofX _M . Every action of ^d by automorphisms of a compact abelian group arises this way. IffR _d , our main formula shows that the topological entropy of is given by

On commuting compact self-adjoint operators on a Pontryagin space

Suppose thatA ₁,A ₂, ...,A _n are compact commuting self-adjoint linear maps on a Pontryagin spaceK of indexk and that their joint root subspaceM ₀ at the zero eigenvalue in ⁿ is a nondegenerate subspace. Then there exist joint invariant subspacesH andF inK such thatK=FH,H is a Hilbert space andF is finite-dimensional space withkdimF(n+2)k. We also consider the structure of restrictionsA _j|F in the casek=1.     

Hausdorff operators on compact abelian groups

Necessary and sufficient conditions are given for the boundedness of Hausdorff operators on the generalized Hardy spaces $H_E^p(G)$, real Hardy space $H_{\mathbb{R}}^1(G)$, $\text{BMO}(G)$, and $\text{BMOA}(G)$ for compact Abelian group G. Surprisingly, these conditions turned out to be the same for all groups and spaces under consideration. Applications to Dirichlet series are given. The case of the space of continuous functions on G and examples are also considered.     

Operators commuting with a discrete subgroup of translations

We study the structure of operators from the Schwartz space S(ℝ ⁿ ) into the tempered distributions S′(ℝ ⁿ ) that commute with a discrete subgroup of translations. The formalism leads to simple derivations of recent results about the frame operator of shift-invariant systems, Gabor, and wavelet frames.     

Spectral analysis for convolution operators on locally compact groups

We consider operators H_μ of convolution with measures μ on locally compact groups. We characterize the spectrum of H_μ by constructing auxiliary operators whose kernel contain the pure point and singular subspaces of H_μ, respectively. The proofs rely on commutator methods.     

Atomization process for convolution operators on locally compact groups

We develop for a large class of locally compact groups a method of approximation of convolution operators on by finitely supported measures with control of the support and of the operator norm of the approximating measures.

     

Manifolds with commuting Jacobi operators

We characterize Riemannian manifolds of constant sectional curvature in terms of commutation properties of their Jacobi operators.