arXiv subjects

Haykel Gaaya

Publications and source records attributed to Haykel Gaaya.

4 recordsLinked to original sources

arXiv2012

A sharpened Schwarz-Pick operatorial inequality for nilpotent operators

Let denote by $S(ϕ)$ the extremal operator defined by the compression of the unilateral shift $S$ to the model subspace $ H(ϕ)=H^{2} \ominus ϕH^{2} $ as the following $S(ϕ)f(z)=P(zf(z)),$ where $P$ denotes the orthogonal projection from the Hardy space $H^{2}$ onto $ H(ϕ)$ and $ϕ$ 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 $S(ϕ)$ in the particular case where $ϕ$ 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 operators

math.FA↗

arXiv2012

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 $\lvert c_{1}\lvert\leqslant c_{0}\cos\fracπ{n+1}$. 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 $ω_{2}(T)\leqslant\cos\fracπ{n+1}$ holds, and $ω_{2}(T)=\cos\fracπ{n+1}$ 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 $\bbs(ϕ)=\bbs_{\lvert Ker(ϕ(\bbs))}$ for a precise inner function $ϕ$. 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 $\bbs(ϕ)$ are given where $ϕ$ is a finite Blashke product with unique zero.

math.FA↗

arXiv2010

On the higher rank numerical range of the shift

For any n-by-n complex matrix T and any $1\leqslant k\leqslant n$, let $Λ_{k}(T)$ the set of all $λ\in \C$ such that $PTP=λP$ 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 $\cos\dfrac{kπ}{n+1}$ if $1<k\leqslant[\frac{n+1}{2}]$ and the empty set if $[\frac{n+1}{2}]<k\leqslant n$, where $[x] $ 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.

math.FA↗

arXiv2010

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 $Λ_{k}(T)$ the set of all $λ\in \C$ such that $PTP=λP$ 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 $\cos\dfrac{kπ}{n+1}$ 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.

math.FA↗