On the second term in the spectral asymptotics of the maxwell operator in a filled resonator

It is shown that the characteristics of the Maxwell operator in a resonator with a smooth inhomogeneous anisotropic filler have constant multiplicity if and only if the matrices of dielectric permittivity and magnetic permeability are connected by the relation ε≡f μ, where f is a scalar-valued function. When ε≡f μ and the boundary is smooth and ideally conducting, the coefficient of λ² in the asymptotic expansion of the distribution function of the eigenvalues of the Maxwell operator turns out to be zero. When the multiplicity of the characteristics is variable, this coefficient can be either zero or nonzero. Bibliography: 22 titles. Translated fromProblemy Matematicheskogo Analiza, No. 13, 1992, pp. 48–79.     

ON THE NON-COMPACTNESS OF THE CORE OF NONATOMIC CONVEX σ-CONTINUOUS MEASURE GAMES

This paper concerns with the core of nonatomic gaxaes of foma f(μ), where μ is a nonatomic nonnegative measure and f is a continuous convex function on the domaln of μ. The main result of this paper is that the core of the game is not compact under the norm topology unless the game itself is a measure. This shows the largeness of the core in a sense other than that defined by Sharky for finite cases.     

On the existence of a Feller semigroup with atomic measure in a nonlocal boundary condition

The existence of Feller semigroups arising in the theory of multidimensional diffusion processes is studied. An elliptic operator of second order is considered on a plane bounded region G. Its domain of definition consists of continuous functions satisfying a nonlocal condition on the boundary of the region. In general, the nonlocal term is an integral of a function over the closure of the region G with respect to a nonnegative Borel measure μ(y, dη) ∈ ∂G. It is proved that the operator is a generator of a Feller semigroup in the case where the measure is atomic. The smallness of the measure is not assumed. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 260, pp. 164–179.     

A quantum model of a real scalar field

A quantum model of a real scalar field with local operator gauge symmetry is discussed. In the localized theory, in order to keep the local operator gauge symmetry, an operator gauge potential BB _μ, is needed. By combining the constraint of operator gauge potentialB _μ, and the microscopic causality theorem, the usual canonical quantization condition of a real scalar field is obtained. Therefore, a quantum model of a real scalar field without the usual procedure of quantizing a related classical model can be directly constructed. Project supported in part by T.D. Lee’s NNSF Grant, National Natural Science Foundation of China, Foundation of Ph. D. Directing Programme of Chinese Universities and the Chinese Academy of Sciences.     

A Spectral Gap Property for Random Walks Under Unitary Representations

Let G be a locally compact group and μ a probability measure on G, which is not assumed to be absolutely continuous with respect to Haar measure. Given a unitary representation $\pi ,\mathcal{H}Let G be a locally compact group and μ a probability measure on G, which is not assumed to be absolutely continuous with respect to Haar measure. Given a unitary representation p,H\pi ,\mathcal{H} of G, we study spectral properties of the operator π(μ) acting on H\mathcal{H} Assume that μ is adapted and that the trivial representation 1 _G is not weakly contained in the tensor product p?[`(p)]\pi\otimes \overline\pi We show that π(μ) has a spectral gap, that is, for the spectral radius r_spec(p(m))r_{\rm spec}(\pi(\mu)) of π(μ), we have r_spec(p(m)) < 1.r_{\rm spec}(\pi(\mu))< 1. This provides a common generalization of several previously known results. Another consequence is that, if G has Kazhdan’s Property (T), then r_spec(p(m)) < 1r_{\rm spec}(\pi(\mu))< 1 for every unitary representation π of G without finite dimensional subrepresentations. Moreover, we give new examples of so-called identity excluding groups.     

Une majoration de la multiplicité spectrale d'opérateurs associés à des cocycles réguliers

We study the spectral multiplicity of unitary operators of defined by cocycles over an irrational rotation α. We prove that the multiplicity is finite whenever the cocycle has bounded variation and we give explicit bounds. For a cocycle given by an absolutely continuous function ϕ on [0,1], we show that the multiplicity is strictly less than max (2.•∫ϕ(x)dx•+1), which is optimal in the case ϕ(x)=nx (where the multiplicity is exactlyn). The proofs are based on the representation of the rotation as a “local rank one” transformation, which arises from the continued fraction expansion of α.      

Invariant subspaces for operators in a general II<Subscript>1</Subscript>-factor

Let ℳ be a von Neumann factor of type II₁ with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ _T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace affiliated with ℳ, such that the Brown measure of is concentrated on B and the Brown measure of is concentrated on ℂ∖B. Moreover, is T-hyperinvariant and the trace of is equal to μ _T(B). In particular, if T∈ℳ has a Brown measure which is not concentrated on a singleton, then there exists a non-trivial, closed, T-hyperinvariant subspace. Furthermore, it is shown that for every T∈ℳ the limit exists in the strong operator topology, and the projection onto is equal to 1_[0,r](A), for every r>0. Supported by The Danish National Research Foundation.     

Spectral multiplicities of infinite measure preserving transformations

Each set E ⊂ ℕ is realized as the set of essential values of the multiplicity function of the Koopman operator for an ergodic conservative infinite measure preserving transformation.     

Expansion of a Self-Adjoint Absolutely Continuous Singular Integral Operator in Generalized Eigenvectors and Its Diagonalization

We describe the relationship between the expansion of a self-adjoint operator in generalized eigenvectors and the direct integral of Hilbert spaces. We perform the explicit diagonalization of a self-adjoint absolutely continuous singular integral operator Y using an Hermitian nonnegative kernel consisting of boundary values of the determining function of the operator T = X + iY with respect to the resolvent of the imaginary part of Y.     

Spectral Analysis of a Transport Operator Arising in Growing Cell Populations

The present paper is concerned with the spectral analysis of a transport-like operator derived from a model introduced by Rotenberg describing the growth of a cell population. Each cell of this population is distinguished by its degree of maturity μ and its maturation velocity v. The biological boundaries of μ = 0 and μ = a (a > 0) are fixed and tightly coupled through mitosis. At mitosis daughter cells and mother cells are related by a general reproduction rule which covers all known biological ones. We first discuss in detail the spectrum of the streaming operator for smooth and partly smooth boundary conditions. Next, we discuss the existence and nonexistence of eigenvalues of the transport operator in the half plane {λ ∈ ℂ : Reλ > where denotes the spectral bound of the streaming operator. In particular, the strict monotonicity of the leading eigenvalue (when it exists) of the transport operator with respect to different parameters of the equation is also considered. We close the paper by describing in detail the various essential spectra of the transport operator for wide classes of collision and boundary operators.     

A Littelmann-type formula for Duistermaat-Heckman measures

Given a compact Hamiltonian T-manifold M, some component of whose moment map is a Palais-Smale Morse function, we give a formula for the Duistermaat-Heckman measure (the Fourier transform of the equivariant symplectic volume) as a sum of volumes of simplices. Unlike previous formulae for this measure, this sum has only positive terms. In the case of M a flag manifold, our results are an asymptotic version of Littelmann's multiplicity result in representation theory [9]. Oblatum 5-I-1998 & 25-I-1998 / Published online: 14 October 1998     

The Markov chain associated to a Pick function

To each function ϕ˜(ω) mapping the upper complex half plane ?⁺ into itself such that the coefficient of ω in the Nevanlinna integral representation is one, we associate the kernel p(y, dx) of a Markov chain on ℝ by

Spectral functions of self-adjoint and symmetric multiplication operators inL 2(X, μ)-spaces

Explicit formulas are derived for the spectral function of double multiplication operator containing a multiplicative evolution inL ²(X, μ)-space and a convolution-type operator inL ²(ℝ ⁿ )-spaces. Symmetric convolution and multiplication operators are considered inL ²(X, μ) andL ²(ℝ ⁿ )-spaces. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 803–810, June, 2000.     

Operator Valued Functions with Pick Operators Having Negative Subspaces of Bounded Dimensions

Functions whose values are bounded linear Hilbert space operators (each operator may be defined on its own subspace of the ambient Hilbert space), the domain of definition is contained in the open unit disc, and having the following property κ, are studied. (κ): All Pick operators associated with the function have the dimensions of their spectral subspace corresponding to the negative part of the spectrum bounded above by a fixed nonnegative integer κ, and the bound κ is attained. No a priori hypotheses concerning regularity of the functions are assumed. A particular class of functions, called standard functions, is introduced, and the corresponding nonnegative integer κ is identified for standard functions. It is proved that every function with property (κ) can be extended to a standard function with property (κ), for the same κ. This result is interpreted as a result on interpolation. As an application, maximal (with respect to the extension relation) functions with the property κ, for a fixed κ, are studied in terms of standard functions. Received: August 5, 2007., Accepted: October 24, 2007.     

Double operator integrals and their estimates in the uniform norm

In this paper, conditions are considered for the existence of the double operator integral ∫∫ ϕ(λ,μ)dE_λTdF_μ, where E_λ, F_μ are the spectral functions of tow self-adjoint operators A, B on a Hilbert space and T is a bounded operator. In principal, the case where A has finite spectrum is studied. Nonlinear estimates of ‖f(A)T-T f(B)‖ in terms of the norm of ‖AT-TB‖ for f∈ Lip 1 are deduced. Also, a formula for the Fréchet derivative is presented. Bibliography: 16 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 148–173. Translated by S. V. Kislyakov.     

Symmetric Ornstein-Uhlenbeck semigroups and their generators

 We provide necessary and sufficient conditions for a Hilbert space-valued Ornstein-Uhlenbeck process to be reversible with respect to its invariant measure μ. For a reversible process the domain of its generator in L ^p (μ) is characterized in terms of appropriate Sobolev spaces thus extending the Meyer equivalence of norms to any symmetric Ornstein-Uhlenbeck operator. We provide also a formula for the size of the spectral gap of the generator. Those results are applied to study the Ornstein-Uhlenbeck process in a chaotic environment. Necessary and sufficient conditions for a transition semigroup (R _t ) to be compact, Hilbert-Schmidt and strong Feller are given in terms of the coefficients of the Ornstein-Uhlenbeck operator. We show also that the existence of spectral gap implies a smoothing property of R _t and provide an estimate for the (appropriately defined) gradient of R _t φ. Finally, in the Hilbert-Schmidt case, we show tha t for any the function R _t φ is an (almost) classical solution of a version of the Kolmogorov equation. Received: 17 September 2001 / Revised version: 3 June 2002 / Published online: 30 September 2002 This work was partially supported by the Small ARC Grant Scheme. Mathematics Subject Classification (2000): Primary: 60H15, 47F05; Secondary: 60J60, 35R15, 35K15 Key words or phrases: Ornstein-Uhlenbeck operator – Second quantization – Reversibility – Spectral gap – Sobolev spaces – Domain of generator     

On order structure and operators in L ∞(μ)

It is known that there is a continuous linear functional on L _∞ which is not narrow. On the other hand, every order-to-norm continuous AM-compact operator from L _∞(μ) to a Banach space is narrow. We study order-to-norm continuous operators acting from L _∞(μ) with a finite atomless measure μ to a Banach space. One of our main results asserts that every order-to-norm continuous operator from L _∞(μ) to c ₀(Γ) is narrow while not every such an operator is AM-compact.     

<Emphasis Type="Italic">BMO</Emphasis>-boundedness of the maximal operator for arbitrary measures

We show that in the one-dimensional case the weighted Hardy-Littlewood maximal operator M _μ is bounded on BMO(μ) for arbitrary Radon measure μ, and that this is not the case in higher dimensions.     

A minimax theorem for irreducible compact operators inL p-spaces

Let (X, Σ, μ) be a σ-finite measure space,T a compact irreducible (positive, linear) operator onL ^p (μ) (1≦p<+∞). It is shown that the spectral radiusr ofT is characterized by the minimax property {fx196-1} where ∑₀ denotes the ring of sets of finite measure and whereQ denotes the set of all, almost everywhere positive functions inL ^p. Moreover, ifr>0 then equality on either side is assumed ifff is the (essentially unique) positive eigenfunction ofT. Various refinements are given in terms of corresponding relations for irreducible finite rank operators approximatingT. Dedicated to H. G. Tillmann on his 60th birthday