A mini-max problem for self-adjoint Toeplitz matrices

We study a minimum problem and associated maximum problem for finite, complex, self-adjoint Toeplitz matrices. If $A$ is such a matrix, of size $(N+1)$-by-$(N+1)$, we identify $A$ with the operator it represents on $P_N$, the space of complex polynomials of degrees at most $N$, with the usual Hilbert space structure it inherits as a subspace of $L^2$ of the unit circle. The operator $A$ is the compression to $P_N$ of the multiplication operator on $L^2$ induced by any function in $L^{\infty}$ whose Fourier coefficients of indices between $-N$ and $N$ match the matrix entries of $A$. Our minimum problem is to minimize the $L^{\infty}$ norm of such inducers. We show there is a unique one of minimum norm, and we describe it. The associated maximum problem asks for the maximum of the ratio of the preceding minimum to the operator norm. That problem remains largely open. We present some suggestive numerical evidence.Comment: 14 pages, new references added, typos fixe

Essential spectra of quasi-parabolic composition operators on Hardy spaces of analytic functions

In this work we study the essential spectra of composition operators on Hardy spaces of analytic functions which might be termed as “quasi-parabolic.” This is the class of composition operators on H2 with symbols whose conjugate with the Cayley transform on the upper half-plane are of the form φ(z)=z+ψ(z), where and (ψ(z))>>0. We especially examine the case where ψ is discontinuous at infinity. A new method is devised to show that this type of composition operator fall in a C*-algebra of Toeplitz operators and Fourier multipliers. This method enables us to provide new examples of essentially normal composition operators and to calculate their essential spectra

Reflexivity of the translation-dilation algebras on L^2(R)

The hyperbolic algebra A_h, studied recently by Katavolos and Power, is the weak star closed operator algebra on L^2(R) generated by H^\infty(R), as multiplication operators, and by the dilation operators V_t, t \geq 0, given by V_t f(x) = e^{t/2} f(e^t x). We show that A_h is a reflexive operator algebra and that the four dimensional manifold Lat A_h (with the natural topology) is the reflexive hull of a natural two dimensional subspace.Comment: 10 pages, no figures To appear in the International Journal of Mathematic

ENTREPRENEURSHIP AND INNOVATION AND SOCIAL ENTREPRENEURSHIP IN VIETNAM: AN EXAMINATION FROM INSTITUTIONAL AND IDEOGRAPHIC LENS

This paper is an introduction to the special issue of Dalat University Journal of Science – Economics and Management on entrepreneurship in Vietnam. There are four papers in this special issue. The first paper examines the impact of institutions on entrepreneurship using data from the Provincial Competitive Index. The second paper utilizes a different set of institutional indicators from the World Bank’s Vietnam Enterprise Survey to assess the impacts of business environment on the development of SMEs. In both papers, the authors find that institutional factors such as entry barriers, lack of policy support systems, informal payment, provincial leadership, lack of access to finance, administrative and procedures, and tax inspections hindered the development of entrepreneurship in Vietnam. The third paper investigates the absence of medium-sized enterprises and the necessity for the development of such enterprises is critically important for Hochiminh City. Using primary and secondary data sources, the author presents a case study on two strategic sectors in the city. The result indicates that medium-sized enterprises are proven to be more effective than large-scale enterprises. The last paper focuses on social entrepreneurship in Vietnam. The authors use Hofstede’s measure of cultural differences to compare social ventures in Vietnam and the United States