论文标题
Tridia-Gonal内核和剩余的可逆操作员，并应用了Aluthge的应用
Tridiagonal kernels and left-invertible operators with applications to Aluthge transforms
论文作者
论文摘要
给定标量$ a_n（\ neq 0）$和$ b_n $，$ n \ geq 0 $，带有带宽$ 1 $的tridiagonal内核或频段内核是开放的单位盘上的积极定义内核$ K $ \ big（（（a_n + b_n z）z^n \ big）\ big（（\ bar {a} _n + \ bar {b} _n \ bar {w}）\ bar {w}^n \ big） \]这定义了一个复制的内核Hilbert空间$ \ MATHCAL {H} _K $（称为tridiagonal Space），分析功能的$ \ Mathbb {d} $，带有$ \ {（a_n + b_nz）z^n \} _ {我们考虑换档运算符$ \ mathcal {h} _k $上的$ m_z $，并证明$ m_z $在且仅当$ \ {| {| {a_n}/{a_n}/{a_ {a_ {n+1}}} | \} | \} | \ e {n \ geq 0} $ bock ake Is vance ake lock ake ake。我们发现，与加权偏移的情况不同，Shimorin的剩余操作员的模型未能将移位的三角形结构带到前景。实际上，当且仅当$ b_0 = 0 $或$ m_z $是加权偏移时，当且仅当shimorin型号下保留了内核$ k $的三角结构。我们证明了有关内核，玻光模型和阳性运算符的三角形不变性的具体分类结果。 我们还开发了一种计算方法来转变变化。奇怪的是，与直接内核空间技术相反，通常，木摩蛋白模型无法产生三角形空间上定义的偏移的三角形Aluthge转换。
Given scalars $a_n (\neq 0)$ and $b_n$, $n \geq 0$, the tridiagonal kernel or band kernel with bandwidth $1$ is the positive definite kernel $k$ on the open unit disc $\mathbb{D}$ defined by \[ k(z, w) = \sum_{n=0}^\infty \Big((a_n + b_n z)z^n\Big) \Big((\bar{a}_n + \bar{b}_n \bar{w}) \bar{w}^n \Big) \qquad (z, w \in \mathbb{D}). \] This defines a reproducing kernel Hilbert space $\mathcal{H}_k$ (known as tridiagonal space) of analytic functions on $\mathbb{D}$ with $\{(a_n + b_nz) z^n\}_{n=0}^\infty$ as an orthonormal basis. We consider shift operators $M_z$ on $\mathcal{H}_k$ and prove that $M_z$ is left-invertible if and only if $\{|{a_n}/{a_{n+1}}|\}_{n\geq 0}$ is bounded away from zero. We find that, unlike the case of weighted shifts, Shimorin's models for left-invertible operators fail to bring to the foreground the tridiagonal structure of shifts. In fact, the tridiagonal structure of a kernel $k$, as above, is preserved under Shimorin model if and only if $b_0=0$ or that $M_z$ is a weighted shift. We prove concrete classification results concerning invariance of tridiagonality of kernels, Shimorin models, and positive operators. We also develop a computational approach to Aluthge transforms of shifts. Curiously, in contrast to direct kernel space techniques, often Shimorin models fails to yield tridiagonal Aluthge transforms of shifts defined on tridiagonal spaces.
