Stein's method and random character ratios

Stein's method is used to prove limit theorems for random character ratios. Tools are developed for four types of structures: finite groups, Gelfand pairs, twisted Gelfand pairs, and association schemes. As one example an error term is obtained for a central limit theorem of Kerov on the spectrum of the Cayley graph of the symmetric group generated by -cycles. Other main examples include an error term for a central limit theorem of Ivanov on character ratios of random projective representations of the symmetric group, and a new central limit theorem for the spectrum of certain random walks on perfect matchings. The results are obtained with very little information: a character formula for a single representation close to the trivial representation and estimates on two step transition probabilities of a random walk. The limit theorems stated in this paper are for normal approximation, but many of the tools developed are applicable for arbitrary distributional approximation.

     

Eigenvalue asymptotics of perturbed periodic Dirac systems in the slow-decay limit

A perturbation decaying to at and not too irregular at introduces at most a discrete set of eigenvalues into the spectral gaps of a one-dimensional Dirac operator on the half-line. We show that the number of these eigenvalues in a compact subset of a gap in the essential spectrum is given by a quasi-semiclassical asymptotic formula in the slow-decay limit, which for power-decaying perturbations is equivalent to the large-coupling limit. This asymptotic behaviour elucidates the origin of the dense point spectrum observed in spherically symmetric, radially periodic three-dimensional Dirac operators.

     

Lyapunov spectra for KMS states on Cuntz-Krieger algebras

We study relations between (H,β)-KMS states on Cuntz-Krieger algebras and the dual of the Perron-Frobenius operator . Generalising the well-studied purely hyperbolic situation, we obtain under mild conditions that for an expansive dynamical system there is a one-one correspondence between (H,β)-KMS states and eigenmeasures of for the eigenvalue 1. We then apply these general results to study multifractal decompositions of limit sets of essentially free Kleinian groups G which may have parabolic elements. We show that for the Cuntz-Krieger algebra arising from G there exists an analytic family of KMS states induced by the Lyapunov spectrum of the analogue of the Bowen-Series map associated with G. Furthermore, we obtain a formula for the Hausdorff dimensions of the restrictions of these KMS states to the set of continuous functions on the limit set of G. If G has no parabolic elements, then this formula can be interpreted as the singularity spectrum of the measure of maximal entropy associated with G. The second author was supported by the DFG project “Ergodentheoretische Methoden in der hyperbolischen Geometrie”.     

FS-property for -algebras

A -algebra is said to have the FS-property if the set of all self-adjoint elements in has a dense subset of elements with finite spectrum. We shall show that this property is not stable under taking the minimal -tensor products even in case of separable nuclear -algebras.

     

Finite simple groups that are not spectrum critical

Let G be a finite group. The spectrum of G is the set ω(G) of orders of all its elements. The subset of prime elements of ω(G) is called the prime spectrum and is denoted by π(G). A group G is called spectrum critical (prime spectrum critical) if, for any subgroups K and L of G such that K is a normal subgroup of L, the equality ω(L/K) = ω(G) (π(L/K) = π(G), respectively) implies that L = G and K = 1. In the present paper, we describe all finite simple groups that are not spectrum critical. In addition, we show that a prime spectrum minimal group G is prime spectrum critical if and only if its Fitting subgroup F(G) is a Hall subgroup of G.     

Single-valued extension property at the points of the approximate point spectrum

A localized version of the single-valued extension property is studied at the points which are not limit points of the approximate point spectrum, as well as of the surjectivity spectrum. In particular, we shall characterize the single-valued extension property at a point in the case that λ_oI−T is of Kato type. From this characterizations we shall deduce several results on cluster points of some distinguished parts of the spectrum.     

Strong duals of projective limits of (LB)-spaces

We investigate the problem when the strong dual of a projective limit of (LB)-spaces coincides with the inductive limit of the strong duals. It is well-known that the answer is affirmative for spectra of Banach spaces if the projective limit is a quasinormable Fréchet space. In that case, the spectrum satisfies a certain condition which is called strong P-type. We provide an example which shows that strong P-type in general does not imply that the strong dual of the projective limit is the inductive limit of the strong duals, but on the other hand we show that this is indeed true if one deals with projective spectra of retractive (LB)-spaces. Finally, we apply our results to a question of Grothendieck about biduals of (LF)-spaces.     

Maximal subalgebras of C⁎-algebras associated with periodic flows

We find necessary and sufficient conditions for the subalgebra of analytic elements associated with a periodic ${\mathrm{C}}^{?}$-dynamical system to be a maximal norm-closed subalgebra. Our conditions are in terms of the Arveson spectrum of the action. We also describe equivalent properties of the system in terms of the strong Connes spectrum and the simplicity of the crossed product.     

Ergodic properties of real cocycles and pseudo-homogeneous Banach spaces

Given an irrational rotation, in the space of real bounded variation functions it is proved that there are ergodic cocycles whose small perturbations remain ergodic; in fact, the set of ergodic cocycles has nonempty dense interior.

Given a pseudo-homogeneous Banach space and an irrational rotation, we study the set of elements satisfying the mean ergodic theorem. Once such a space is not homogeneous, we prove it is not reflexive and not separable. In ``natural" cases, up to -cohomology, the only elements satisfying the mean ergodic theorem are those from the closure of trigonometric polynomials.

For pseudo-homogeneous spaces admitting a Koksma's inequality ergodicity of the corresponding cylinder flows can be deduced from spectral properties of some circle extensions. In particular this is the case of Lebesgue spectrum (in the orthocomplement of the space of eigenfunctions) for the circle extension.

     

Extremal vectors and invariant subspaces

For a bounded linear operator on Hilbert space we define a sequence of so-called minimal vectors in connection with invariant subspaces and show that this presents a new approach to invariant subspaces. In particular, we show that for any compact operator some weak limit of the sequence of minimal vectors is noncyclic for all operators commuting with and that for any normal operator , the norm limit of the sequence of minimal vectors is noncyclic for all operators commuting with . Thus, we give a new and more constructive proof of existence of invariant subspaces. The sequence of minimal vectors does not seem to converge in norm for an arbitrary bounded linear operator. We will prove that if belongs to a certain class of operators, then the sequence of such vectors converges in norm, and that if belongs to a subclass of , then the norm limit is cyclic.

     

The stability radius of a quasi-Fredholm operator

We extend the technique used by Kordula and Müller to show that the stability radius of a quasi-Fredholm operator is the limit of as . If is an isolated point of the Apostol spectrum , then the above limit is non-zero if and only if is quasi-Fredholm.

     

Aur la vacuité du spectre d'un élément d'une algèbre\mathcal{L}\mathcal{F}

We give an example of a complete commutative unitary and semi-simple topological algebra, which is a locally convex inductive limit of an increasing sequence of Fréchet algebras ( algebra), and which contains the field (X) of rational functions; so it contains elements which have empty spectrum and therefore does not contain any character, neither continuous nor non-continuous. This unitary algebra is not a division algebra, so it contains at least one non-trivial maximal ideal; but none of its maximal ideals is closed and they all have infinite codimension. The Gelfand-Mazur Theorem remains therefore unknown for algebras.

     

The homogeneous spectrum of a graded commutative ring

Suppose is a torsion-free cancellative commutative monoid for which the group of quotients is finitely generated. We prove that the spectrum of a -graded commutative ring is Noetherian if its homogeneous spectrum is Noetherian, thus answering a question of David Rush. Suppose is a commutative ring having Noetherian spectrum. We determine conditions in order that the monoid ring have Noetherian spectrum. If , we show that has Noetherian spectrum, while for each we establish existence of an example where the homogeneous spectrum of is not Noetherian.

     

Linear resolvent growth of rank one perturbation of a unitary operator does not imply its similarity to a normal operator

The main result of this paper is that for any unitary (selfadjoint) operatorU with non-trivial absolutely continuous part of the spectrum, there exists a rank-one perturbationK = ba* = (a)b such that the operatorT = U + K satisfies the Linear Resolvent Growth condition (LRG),

All non-P-points are the limits of nontrivial sequences in supercompact spaces

A Hausdorff topological space is called supercompact if there exists a subbase such that every cover consisting of this subbase has a subcover consisting of two elements. In this paper, we prove that every non-P-point in any continuous image of a supercompact space is the limit of a nontrivial sequence. We also prove that every non-P-point in a closed -subspace of a supercompact space is a cluster point of a subset with cardinal number But we do not know whether this statement holds when replacing by the countable cardinal number. As an application, we prove in ZFC that there exists a countable stratifiable space which has no supercompact compactification.

     

On Krein's example

In his 1953 paper [Matem. Sbornik 33 (1953), 597-626] Mark Krein presented an example of a symmetric rank one perturbation of a self-adjoint operator such that for all values of the spectral parameter in the interior of the spectrum, the difference of the corresponding spectral projections is not trace class. In the present note it is shown that in the case in question this difference has simple Lebesgue spectrum filling in the interval and, therefore, the pair of the spectral projections is generic in the sense of Halmos but not Fredholm.

     

Another note on Weyl's theorem

``Weyl's theorem holds" for an operator on a Banach space when the complement in the spectrum of the ``Weyl spectrum" coincides with the isolated points of spectrum which are eigenvalues of finite multiplicity. This is close to, but not quite the same as, equality between the Weyl spectrum and the ``Browder spectrum", which in turn ought to, but does not, guarantee the spectral mapping theorem for the Weyl spectrum of polynomials in . In this note we try to explore these distinctions.

     

Isolated spectral points

The paper studies isolated spectral points of elements of Banach algebras and of bounded linear operators in terms of the existence of idempotents, and gives an elementary characterization of spectral idempotents. It is shown that is isolated in the spectrum of a bounded linear operator if the (not necessarily closed) space is nonzero and complemented by a closed subspace satisfying .

     

Essentially Normal Operator + Compact Operator = Strongly Irreducible Operator

It is shown that given an essentially normal operator with connected spectrum, there exists a compact operator such that is strongly irreducible.