论文标题
在伯格曼空间上紧凑型toeplitz操作员的光谱特性，具有对数腐烂的符号和对带矩阵的应用
On spectral properties of compact Toeplitz operators on Bergman space with logarithmically decaying symbol and applications to banded matrices
论文作者
论文摘要
令$ l^2（d）$为单元磁盘上可测量的正方形功能的空间。令$ l^2_a（d）$为Bergman Space，即$ l^2（d）$中的分析功能的（封闭）子空间。 $ p _+$ $ $用于正交投影，从$ l^2（d）$到$ l^2_a（d）$。对于功能$φ\在l^\ infty（d）$中，toeplitz运算符$t_φ：l^2_a（d）\ to l^2_a（d）$定义为$$ t_或t_φf= p_+φf，\ quad f \ quad f \ quad f \ in l^2_a（d）。 $$本文的主要结果是光谱渐近学用于带有对数衰减符号的紧凑型toeplitz运算符的奇异（或特征）值，即$$φ（z）=φ_1（e^{i^{iθ}}） $ z = re^{iθ} $和$φ_1$是单位圆上的连续（或零件连续）功能。结果应用于带（包括雅各比）矩阵的光谱分析。
Let $L^2(D)$ be the space of measurable square-summable functions on the unit disk. Let $L^2_a(D)$ be the Bergman space, i.e., the (closed) subspace of analytic functions in $L^2(D)$. $P_+$ stays for the orthogonal projection going from $L^2(D)$ to $L^2_a(D)$. For a function $φ\in L^\infty(D)$, the Toeplitz operator $T_φ: L^2_a(D)\to L^2_a(D)$ is defined as $$ T_φf=P_+φf, \quad f\in L^2_a(D). $$ The main result of this article are spectral asymptotics for singular (or eigen-) values of compact Toeplitz operators with logarithmically decaying symbols, that is $$ φ(z)=φ_1(e^{iθ})\, (1+\log(1/(1-r)))^{-γ},\quad γ>0, $$ where $z=re^{iθ}$ and $φ_1$ is a continuous (or piece-wise continuous) function on the unit circle. The result is applied to the spectral analysis of banded (including Jacobi) matrices.
