Journal of the Australian Mathematical Society

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Volume 98 - June 2015

Contents

Page 2 of 2

LOGARITHMIC CONVEXITY OF AREA INTEGRAL MEANS FOR ANALYTIC FUNCTIONS II
Part of:
- Spaces and algebras of analytic functions
CHUNJIE WANG, JIE XIAO, KEHE ZHU
Published online by Cambridge University Press:

14 October 2014, pp. 117-128
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For $0<p<\infty$ and $-2\leq {\it\alpha}\leq 0$ we show that the $L^{p}$ integral mean on $r\mathbb{D}$ of an analytic function in the unit disk $\mathbb{D}$ with respect to the weighted area measure $(1-|z|^{2})^{{\it\alpha}}\,dA(z)$ is a logarithmically convex function of $r$ on $(0,1)$.

AN INTEGRAL REPRESENTATION FOR BESOV AND LIPSCHITZ SPACES
Part of:
- Holomorphic functions of several complex variables
- Spaces and algebras of analytic functions
KEHE ZHU
Published online by Cambridge University Press:

17 October 2014, pp. 129-144
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It is well known that functions in the analytic Besov space $B_{1}$ on the unit disk $\mathbb{D}$ admit an integral representation
$$\begin{eqnarray}f(z)=\int _{\mathbb{D}}\frac{z-w}{1-z\overline{w}}\,d{\it\mu}(w),\end{eqnarray}$$
where ${\it\mu}$ is a complex Borel measure with $|{\it\mu}|(\mathbb{D})<\infty$. We generalize this result to all Besov spaces $B_{p}$ with $0<p\leq 1$ and all Lipschitz spaces ${\rm\Lambda}_{t}$ with $t>1$. We also obtain a version for Bergman and Fock spaces.

LINEAR MAPS PRESERVING TENSOR PRODUCTS OF RANK-ONE HERMITIAN MATRICES
Part of:
- Special matrices
- Basic linear algebra
JINLI XU, BAODONG ZHENG, AJDA FOŠNER
Published online by Cambridge University Press:

21 November 2014, pp. 407-428
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For a positive integer $n\geq 2$, let $M_{n}$ be the set of $n\times n$ complex matrices and $H_{n}$ the set of Hermitian matrices in $M_{n}$. We characterize injective linear maps ${\it\phi}:H_{m_{1}\cdots m_{l}}\rightarrow H_{n}$ satisfying
$$\begin{eqnarray}\text{rank}(A_{1}\otimes \cdots \otimes A_{l})=1\Longrightarrow \text{rank}({\it\phi}(A_{1}\otimes \cdots \otimes A_{l}))=1\end{eqnarray}$$
for all $A_{k}\in H_{m_{k}}$, $k=1,\dots ,l$, where $l,m_{1},\dots ,m_{l}\geq 2$ are positive integers. The necessity of the injectivity assumption is shown. Moreover, the connection of the problem to quantum information science is mentioned.