Reduced spectral synthesis and compact operator synthesis

We introduce and study the notion of reduced spectral synthesis, which unifies the concepts of spectral synthesis and uniqueness in locally compact groups. We exhibit a number of examples and prove that every non-discrete locally compact group with an open abelian subgroup has a subset that fails reduced spectral synthesis. We introduce compact operator synthesis as an operator algebraic counterpart of this notion and link it with other exceptional sets in operator algebra theory, studied previously. We show that a closed subset $E$ of a second countable locally compact group $G$ satisfies reduced local spectral synthesis if and only if the subset $E^* = \{(s,t) : ts^{-1}\in E\}$ of $G\times G$ satisfies compact operator synthesis. We apply our results to questions about the equivalence of linear operator equations with normal commuting coefficients on Schatten $p$-classes.Comment: 43 page

Closable Multipliers

Let (X,m) and (Y,n) be standard measure spaces. A function f in $L^\infty(X\times Y,m\times n)$ is called a (measurable) Schur multiplier if the map $S_f$, defined on the space of Hilbert-Schmidt operators from $L_2(X,m)$ to $L_2(Y,n)$ by multiplying their integral kernels by f, is bounded in the operator norm. The paper studies measurable functions f for which $S_f$ is closable in the norm topology or in the weak* topology. We obtain a characterisation of w*-closable multipliers and relate the question about norm closability to the theory of operator synthesis. We also study multipliers of two special types: if f is of Toeplitz type, that is, if f(x,y)=h(x-y), x,y in G, where G is a locally compact abelian group, then the closability of f is related to the local inclusion of h in the Fourier algebra A(G) of G. If f is a divided difference, that is, a function of the form (h(x)-h(y))/(x-y), then its closability is related to the "operator smoothness" of the function h. A number of examples of non-closable, norm closable and w*-closable multipliers are presented.Comment: 35 page

Secondary electron emission from sodium chloride, glass and aluminum oxide at various temperature

The method of single impulses was used to measure the coefficients of the secondary electronic emission for 2 types of Al2O2, monocrystalline NaCl and glass at different temperatures and for different values of the energy of the primary electrons. The value of the secondary electron emission does not depend upon temperature. The effect of a gas film on the value of the secondary electron emission was detected

Optimization Approach to the Treatment of Open Boundary-Conditions

A solution to an optimization problem is developed that deals with minimizing a measure of difference between the values of observed and predicted variables at an open ocean boundary. Minimization is based on the change of the flux of energy through the open boundary. It is shown that many of the longwave radiation conditions that are commonly used in ocean modeling can be derived using this optimization criteria. However, the minimization process is seen to produce a modification of these radiation conditions in that they are multiplied by a coefficient, which allows the conditions to adapt to a change in the flux of energy penetrating the boundary. An example of the numerical implementation is presented for the Reid and Bodine boundary formulation. For a standing wave problem with an analytical solution, use of the modified Reid and Bodine formulation is seen to eliminate almost entirely errors in the predicted amplitudes and phases. Overall, this approach is seen to allow a modeler to generate different types of boundary conditions based on observations as well as the inclinations of the modeler

Eye position modulates retinotopic responses in early visual areas: a bias for the straight-ahead direction

Even though the eyes constantly change position, the location of a stimulus can be accurately represented by a population of neurons with retinotopic receptive fields modulated by eye position gain fields. Recent electrophysiological studies, however, indicate that eye position gain fields may serve an additional function since they have a non-uniform spatial distribution that increases the neural response to stimuli in the straight-ahead direction. We used functional magnetic resonance imaging and a wide-field stimulus display to determine whether gaze modulations in early human visual cortex enhance the blood-oxygenation-level dependent (BOLD) response to stimuli that are straight-ahead. Subjects viewed rotating polar angle wedge stimuli centered straight-ahead or vertically displaced by ±20° eccentricity. Gaze position did not affect the topography of polar phase-angle maps, confirming that coding was retinotopic, but did affect the amplitude of the BOLD response, consistent with a gain field. In agreement with recent electrophysiological studies, BOLD responses in V1 and V2 to a wedge stimulus at a fixed retinal locus decreased when the wedge location in head-centered coordinates was farther from the straight-ahead direction. We conclude that stimulus-evoked BOLD signals are modulated by a systematic, non-uniform distribution of eye-position gain fields

Some remarks on invariant subspaces in real Banach spaces (revised version)

It is proved that a commutative algebra $A$ of operators on a reflexive real Banach space has an invariant subspace if each operator $T\in A$ satisfies the condition $$\|1- \varepsilon T^2\|_e \le 1 + o(\varepsilon) \text{ when } \varepsilon\searrow 0,$$ where $\|\cdot\|_e$ is the essential norm. This implies the existence of an invariant subspace for every commutative family of essentially selfadjoint operators on a real Hilbert space

Quenching of dynamic nuclear polarization by spin-orbit coupling in GaAs quantum dots

The central-spin problem, in which an electron spin interacts with a nuclear spin bath, is a widely studied model of quantum decoherence. Dynamic nuclear polarization (DNP) occurs in central spin systems when electronic angular momentum is transferred to nuclear spins and is exploited in spin-based quantum information processing for coherent electron and nuclear spin control. However, the mechanisms limiting DNP remain only partially understood. Here, we show that spin-orbit coupling quenches DNP in a GaAs double quantum dot, even though spin-orbit coupling in GaAs is weak. Using Landau-Zener sweeps, we measure the dependence of the electron spin-flip probability on the strength and direction of in-plane magnetic field, allowing us to distinguish effects of the spin-orbit and hyperfine interactions. To confirm our interpretation, we measure high-bandwidth correlations in the electron spin-flip probability and attain results consistent with a significant spin-orbit contribution. We observe that DNP is quenched when the spin-orbit component exceeds the hyperfine, in agreement with a theoretical model. Our results shed new light on the surprising competition between the spin-orbit and hyperfine interactions in central-spin systems.Comment: 5+12 pages, 9 figure

Reduced synthesis in harmonic analysis and compact synthesis in operator theory

The notion of reduced synthesis in the context of harmonic analysis on general locally compact groups is introduced; in the classical situation of commutative groups, this notion means that a function f in the Fourier algebra is annihilated by any pseudofunction supported on f −1(0). A relationship between reduced synthesis and compact synthesis (i.e., the possibility of approximating compact operators by pseudointegral ones without increasing the support) is determined, which makes it possible to obtain new results both in operator theory and in harmonic analysis. Applications to the theory of linear operator equations are also given