A sharpened Schwarz-Pick operatorial inequality for nilpotent operators

Let denote by $S(\phi)$ the extremal operator defined by the compression of the unilateral shift $S$ to the model subspace $H(\phi)=H^{2} \ominus \phi H^{2}$ as the following $S(\phi)f(z)=P(zf(z)),$ where $P$ denotes the orthogonal projection from the Hardy space $H^{2}$ onto $H(\phi)$ and $\phi$ is an inner function on the unit disc. In this mathematical notes, we give an explicit formula of the numerical radius of the truncated shift $S(\phi)$ in the particular case where $\phi$ is a finite Blaschke product with unique zero and an estimate on the general case. We establish also a sharpened Schwarz-Pick operatorial inequality generalizing a U. Haagerup and P. de la Harpe result for nilpotent operator

On the numerical radius of the truncated adjoint Shift

A celebrated thorem of Fejer (1915) asserts that for a given positive trigonometric polynomial $\sum_{j=-n+1}^{n-1}c_{j}e^{ijt}$, we have $\lvert c_{1}\lvert\leqslant c_{0}\cos\frac{\pi}{n+1}$. A more recent inequality due to U. Haagerup and P. de la Harpe asserts that, for any contraction $T$ such that $T^{n}=0$, for some $n\geq2$, the inequality $\omega_{2}(T)\leqslant\cos\frac{\pi}{n+1}$ holds, and $\omega_{2}(T)=\cos\frac{\pi}{n+1}$ when T is unitarily equivalent to the extremal operator {S}^{\ast}_{n}={\bbs}_{\lvert{\C}^{n}}={\bbs}_{\lvert Ker (u_{n}(\bbs))} where $u_{n}(z)=z^{n}$ and \bbs is the adjoint of the shift operator on the Hilbert space of all square summable sequences. Apparently there is no relationship between them. In this mathematical note, we show that there is a connection between Taylor coefficients of positive rational functions on the torus and numerical radius of the extremal operator \bbs(\phi)=\bbs_{\lvert Ker(\phi(\bbs))} for a precise inner function $\phi$. This result completes a line of investigation begun in 2002 by C. Badea and G. Cassier \cite{Cassier}. An upper and lower bound of the numerical radius of \bbs(\phi) are given where $\phi$ is a finite Blashke product with unique zero

On the higher rank numerical range of the shift operator

For any n-by-n complex matrix T and any $1\leqslant k\leqslant n$, let $\Lambda_{k}(T)$ the set of all \lambda\in \C such that $PTP=\lambda P$ for some rank-k orthogonal projection $P$ be its higher rank-k numerical range. It is shown that if \bbS is the n-dimensional shift on {\C}^{n} then its rank-k numerical range is the circular disc centred in zero and with radius $\cos\dfrac{k\pi}{n+1}$ if $1<k\leqslant\left[\frac{n+1}{2} \right]$ and the empty set if $\left[\frac{n+1}{2} \right]<k\leqslant n$, where $\left[x \right]$ denote the integer part of $x$. This extends and rafines previous results of U. Haagerup, P. de la Harpe \cite{Haagerup} on the classical numerical range of the n-dimensional shift on{\C}^{n}. An interesting result for higher rank-$k$ numerical range of nilpotent operator is also established

The Macroeconomic Social Accounting Matrix of Tunisia in 1996

This paper shows the characteristics of the macroeconomic Social Accounting Matrix (SAM) of Tunisia in 1996. It is established that the SAM is one of the data bases of the Computable General Equilibrium (CGE) Models. These last years, the researchers have intensified the use of the SAM within the framework of the modelling of CGE within their work. The SAM is regarded as the general data base because it contains the whole relationship between the economic agents of a given economy, including the existing relationship on the level of table input – output. The SAM can be presented in two forms: the aggregate form (or macroeconomic) and desegregated form (or microeconomic). In this paper, we will focus on the first form of SAM corresponding to Tunisia during 1996. This study is made up of two sections. In the first section, we will present the construction of the unbalanced Macroeconomic Social Accounting Matrix (SAMmac) of Tunisia in 1996. This section includes three sub-sections. The first sub-section contains the accounts of our SAMmac in 1996. Then, the second sub-section will be devoted to the statistical sources used for the construction of our SAMmac. Finally, the last sub-section will concentrate on our SAMmac which is in imbalance because of the various sources used. We present, in the second section, the balancing of SAMmac of Tunisia in 1996 in basing upon the method known as ' Cross-Entropy '. Initially, in a first sub-section, we will define the ' Cross-Entropy ' approach. Then, in a second sub- section, we will use this last approach to balance our SAMmac. Conclusions follow.Social Accounting Matrix, Cross-Entropy, Tunisia, balanced of the SAM

Various methods of balancing of the macro SAM of Tunisia during the year 2000

This paper has the aim of presenting the various methods of balancing applied to the macroeconomic Social Accounting Matrix (macro SAM) of Tunisia during the year 2000. These methods (method of entropy, method of least squares....) were used by the modellers of Computable General Equilibrium (CGE) whose would to balance the totals in columns and the totals in lines of the SAM of the developing countries in question. To be able to be regarded as the base of data of a CGE model, the matrix must check the principle of balance of the totals in columns and in lines of each one of these accounts.

Impacts of the Euro-Tunisian agreements of free exchange: evaluation by a Computable General Equilibrium Model in 1996

Until nowadays, the South-Mediterranean countries have focused on their strategic choice which consists in the creation of a Euro-Mediterranean Free Trade Area (FTA). Among the first participants in the constitution of this zone, is Tunisia, which signed an agreement of association with the European Union (EU). Since the signature of this agreement, Tunisia has been interested in improving its internal economic situation through internal reforms to facilitate the application of external reforms. With the coming into force of this agreement, the free euro-Tunisian exchange concerns exclusively industrial products, while the other products (farm produce) are going to be examined in the coming days. The object of this communication is to compare the effects of the free total exchange and the effects of the free industrial exchange between Tunisia and the EU in the Tunisian economy. This problem was estimated by means of a Computable General Equilibrium Model (CGEM). This document is going to be divided in two blocks of simulations. The first concerns total commercial liberalisation, while the second block consists in a partial commercial liberalisation. We notice that a progressive commercial liberalisation applied during a reasonable period remains an effective economic reform for a developing country. Moreover, such a liberalisation is considered as preferable when it is applied to one or to a certain category of product. In other words, free progressive industrial exchange between Tunisia and the EU is considered as a good strategy for the constitution of a Euro-Mediterranean FTA on the horizon 2010.CGEM, trade policy, Tunisia, trade negotiations, simulations