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COMPOSITION OPERATORS ON UNIFORM ALGEBRAS AND THE PSEUDOHYPERBOLIC METRIC
Galindo, P.,Gamelin, T.W.,Lindstrom, M. Korean Mathematical Society 2004 대한수학회지 Vol.41 No.1
Let A be a uniform algebra, and let $\phi$ be a self-map of the spectrum $M_A$ of A that induces a composition operator $C_{\phi}$, on A. It is shown that the image of $M_A$ under some iterate ${\phi}^n$ of \phi is hyperbolically bounded if and only if \phi has a finite number of attracting cycles to which the iterates of $\phi$ converge. On the other hand, the image of the spectrum of A under $\phi$ is not hyperbolically bounded if and only if there is a subspace of $A^{**}$ "almost" isometric to ${\ell}_{\infty}$ on which ${C_{\phi}}^{**}$ "almost" an isometry. A corollary of these characterizations is that if $C_{\phi}$ is weakly compact, and if the spectrum of A is connected, then $\phi$ has a unique fixed point, to which the iterates of $\phi$ converge. The corresponding theorem for compact composition operators was proved in 1980 by H. Kamowitz [17].
Composition operators on uniform algebras and the pseudohyperbolic metric
P. Galindo,T. W. Gamelin,M. Lindstr\ 대한수학회 2004 대한수학회지 Vol.41 No.1
Let A be a uniform algebra, and let φ be a self-map of the spectrum MA of A that induces a composition operator Cφ on A. It is shown that the image of MA under some iterate φn of φ is hyperbolically bounded if and only if φ has a finite number of attracting cycles to which the iterates of φ converge. On the other hand, the image of the spectrum of A under φ is not hyperbolically bounded if and only if there is a subspace of A∗∗ “almost” isometric to ∞ on which C∗∗ φ is “almost” an isometry. A corollary of these characterizations is that if Cφ is weakly compact, and if the spectrum of A is connected, then φ has a unique fixed point, to which the iterates of φ converge. The corresponding theorem for compact composition operators was proved in 1980 by H. Kamowitz [17].
Magnetic Polaron Formation Dynamics in Mn^(2+)-Doped Colloidal Nanocrystals up to Room Temperature
Gerd Bacher,Lars Schneider,Rémi Beaulac,Paul I. Archer,Daniel R. Gamelin 한국물리학회 2011 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.58 No.51
The dynamics of exciton magnetic polaron (EMP) formation after pulsed laser excitation is studied in Mn^(2+)-doped colloidal CdSe nanocrystals. A transient energy shift of more than 130 meV at low temperature is obtained for the characteristic photoluminescence signal, and is related to the alignment of the Mn^(2+)-ion spins in the exchange field of the optically generated electron-hole pairs. The dynamics of laser induced magnetization are characterized by at least two time constants attributed to the EMP formation within a few hundreds of picoseconds, followed by a subsequent reorientation of the EMP spin complex in the nanosecond regime. We found a complete spin alignment up to temperatures of about 50 K and clear signatures of optically induced magnetization up to room temperature.