The Carath\'eodory-Fej\'er Interpolation Problems and the von-Neumann Inequality

The validity of the von-Neumann inequality for commuting $n$ - tuples of $3\times 3$ matrices remains open for $n\geq 3$. We give a partial answer to this question, which is used to obtain a necessary condition for the Carath\'{e}odory-Fej\'{e}r interpolation problem on the polydisc $\mathbb D^n.$ In the special case of $n=2$ (which follows from Ando's theorem as well), this necessary condition is made explicit. An alternative approach to the Carath\'{e}odory-Fej\'{e}r interpolation problem, in the special case of $n=2,$ adapting a theorem of Kor\'{a}nyi and Puk\'{a}nzsky is given. As a consequence, a class of polynomials are isolated for which a complete solution to the Carath\'{e}odory-Fej\'{e}r interpolation problem is easily obtained. A natural generalization of the Hankel operators on the Hardy space of $H^2(\mathbb T^2)$ then becomes apparent. Many of our results remain valid for any $n\in \mathbb N,$ however, the computations are somewhat cumbersome for $n>2$ and are omitted. The inequality $\lim_{n\to \infty}C_2(n)\leq 2 K^\mathbb C_G$, where $K_G^\mathbb C$ is the complex Grothendieck constant and $$C_2(n)=\sup\big\{\|p(\boldsymbol T)\|:\|p\|_{\mathbb D^n,\infty}\leq 1, \|\boldsymbol T\|_{\infty} \leq 1 \big\}$$ is due to Varopoulos. Here the supremum is taken over all complex polynomials $p$ in $n$ variables of degree at most $2$ and commuting $n$ - tuples $\boldsymbol T:=(T_1,\ldots,T_n)$ of contractions. We show that $$\lim_{n\to \infty}C_2(n)\leq \frac{3\sqrt{3}}{4} K^\mathbb C_G$$ obtaining a slight improvement in the inequality of Varopoulos. We show that the normed linear space $\ell^1(n),$ $n>1,$ has no isometric embedding into $k\times k$ complex matrices for any $k\in \mathbb N$ and discuss several infinite dimensional operator space structures on it.Comment: This is my thesis submitted to Indian Institute of Science, Bangalore on 20th July, 201

Spin Glass-like Phase below ~ 210 K in Magnetoelectric Gallium Ferrite

In this letter we show the presence of a spin-glass like phase in single crystals of magnetoelectric gallium ferrite (GaFeO3) below ~210 K via temperature dependent ac and dc magnetization studies. Analysis of frequency dispersion of the susceptibility peak at ~210 K using the critical slowing down model and Vogel-Fulcher law strongly suggests the existence of a classical spin-glass like phase. This classical spin glass behavior of GaFeO3 is understood in terms of an outcome of geometrical frustration arising from the inherent site disorder among the antiferromagnetically coupled Fe ions located at octahedral Ga and Fe sites.Comment: 16 pages, 4 figure

Origin of the unusual dependence of Raman D band on excitation wavelength in graphite-like materials

We have revisited the still unresolved puzzle of the dispersion of the Raman disordered-induced D band as a function of laser excitation photon energy E$_L$ in graphite-like materials. We propose that the D-mode is a combination of an optic phonon at the K-point in the Brillioun zone and an acoustic phonon whose momentum is determined uniquely by the double resonance condition. The fit of the experimental data with the double-resonance model yields the reduced effective mass of 0.025m$_{e}$ for the electron-hole pairs corresponding to the A$_{2}$ transition, in agreement with other experiments. The model can also explain the difference between $\omega_S$ and $\omega_{AS}$ for D and D$^{\star}$ modes, and predicts its dependence on the Raman excitation frequency.Comment: 4 figures in eps forma