Concrete model theory for a class of operators with unitary part

Let m and v_t, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral $\text{H}$ = ⊕L²(v^t) dm(t) and the operator $\text{(L?)(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{?(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∫}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∫}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ on $\text{H}$, where e(s, t) = exp ∫_s^t ∫_Tdv_λ(θ) dm(λ). Let μ_t be the measure defined by $\text{∫}{}_{\mathrm{T}}\text{?(x) dμ}{}_{\mathrm{t}}\text{(x) = ∫}{}_0{}^{\mathrm{t}}\text{∫}{}_{\mathrm{T}}\text{?(x) dv}{}_{\mathrm{s}}\text{dm(s)}$ for all continuous ?, and let ?_t(z) = exp[?∫ (e^iθ + z)(e^iθ ? z)^?1dμ_t(gq)]. Call {v_t} regular iff for all $\text{t, ¦?}{}_{\mathrm{t}}\text{(e}{}^{\mathrm{i\theta }}\text{)¦ = ¦?}{}_{\mathrm{2\pi }}\text{(e}{}^{\mathrm{i\theta }}\text{)¦}$ for 1 a.e.     

Unitary point spectrum of almost unitary operators

In order that a set, be the unitary point spectrum of some almost unitary operator (i.e., of an operator of the form U +K, where U is a unitary operator and K₁) it is necessary and sufficient that E be a countable union of Carleson sets.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 143–149, 1983.     

Measure-compact operators, almost compact operators, and linear functional equations in L p

Under study are the measure-compact operators and almost compact operators in L _p . We construct an example of a measure-compact operator that is not almost compact. Introducing two classes of closed linear operators in L _p , we prove that the resolvents of these operators are almost compact or measure-compact. We present methods for the reduction of linear functional equations of the second kind in L _p with almost compact or measure-compact operators to equivalent linear integral equations in L _p with quasidegenerate Carleman kernels.     

Scattering theory for elliptic operators of arbitrary order

We prove the main theorems of scattering theory for selfadjoint elliptic partial differential operators of arbitrary order. Under various hypotheses we show that the wave operators exist and are complete, that the intertwining relations hold, and that the invariance principle holds. One of our main hypotheses is that each lower order coefficientq(x) satisfies. $$(1 + \left| x \right|)^\alpha \int\limits_{\left| {x - y} \right|< a} {\left| {q(y)} \right|dy \in L^p (E^n )}$$ for some α≥0,a>0 and forp≤∞ such that $$\alpha > 1 - \frac{{2n}}{{(n + 1)p}}$$      

Wave operators for non-self-adjoint operators and the functional model

Explicit formulas for the wave operators satisfying certain conditions are obtained in terms of the functional model of Nagy-Foias.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 69, pp. 129–135, 1977.The author is grateful to B. S. Pavlov for his attention to this work.     

Concrete model theory for a class of operators

The operator $\text{L?(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∝}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∝}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ acting on H=∝₀^2π⊕L²(v_t), where m and v_t, 0 ? t ? 2π are measures on [0, 2π] with m smooth and e(s, t) = exp[?∝_t^s∝_Tdv_λ(θ) dm(λ)], satisfies $\text{rank}\text{(I ? LL}{}^1\text{) =}\text{rank}\text{(I ? L}{}^1\text{L) = 1}$. It is, therefore, unitarily equivalent to a scalar Sz.-Nagy-Foia? canonical model. The purpose of this paper is to determine the model explicitly and to give a formula for the unitary equivalence.     

Towards a model theory for 2-hyponormal operators

We introduce the notion ofweak subnormality, which generalizes subnormality in the sense that for the extension ofT we only require that hold forf ; in this case we call a partially normal extension ofT. After establishing some basic results about weak subnormality (including those dealing with the notion of minimal partially normal extension), we proceed to characterize weak subnormality for weighted shifts and to prove that 2-hyponormal weighted shifts are weakly subnormal. Let { _n } _n=0 be a weight sequence and letW denote the associated unilateral weighted shift on . IfW is 2-hyponormal thenW is weakly subnormal. Moreover, there exists a partially normal extension on such that (i) is hyponormal; (ii) ; and (iii) . In particular, if is strictly increasing then can be obtained as

Trace formulas for almost commuting operators,cyclic cohomology and subnormal operators

Some formulas related to cyclic cohomology for the trace of the product of commutators are established. A simple complete unitary invariant for some subnormal operator with simply connected spectrum is found.This work is supported in part by NSF grant.     

The relationship between almost Dunford-Pettis operators and almost limited operators

We investigate Banach lattices on which each positive almost Dunford- Pettis operator is almost limited and conversely.     

Spectral theory of difference operators with almost constant coefficients

The spectrum of higher even order difference operators with almost constant coefficients is determined. With appropriate smoothness and decay conditions on the coefficients, we show that singular continuous spectrum is absent and that the absolutely continuous spectrum agrees with that of the constant coefficient limiting operator. For such operators, the absolutely continuous spectrum is determined uniquely by the range of the characteristic polynomial. This result extends a similar result for even order differential operators. The methods of proof are closely related likewise. Finally, some results on fourth order operators with unbounded coefficients are shown.     

OnL-functions and intertwining operators for unitary groups

The ingredients of an “L-function machine” for the quasi-split groupU _{n, n} +1 × Res GL _n are treated here, following similar theories of P. Shapiro and S. Gelbart. We start with a known Rankin-Selberg type integral having an Euler product. In section 2 we compute the local integral to get a localL function. This is done by working with an “L group” related to ^L G and the relative root system. All computations are carried out for the split and the non-split case. In section 3 we address the problem of analytic continuation of the Eisenstein series. This involves computation of poles of intertwining operators.     

Spectral theory of difference operators with almost constant coefficients II

We show that the operators whose coefficients are approximately constant in a general sense have an absolutely continuous spectrum which is equal to that of the corresponding constant coefficient operator. For such operators, the absolutely continuous spectrum can be read off from the associated characteristic polynomial. This generalizes the classical results on second-order operators and extends those of higher order differential operators to the difference setting. Our approach relies on an analysis of the associated difference equation with the help of uniform asymptotic summation techniques.