论文标题
广义球形Aluthge变换的光谱图和关节光谱半径
The Spectral Picture and Joint Spectral Radius of the Generalized Spherical Aluthge Transform
论文作者
论文摘要
对于任意通勤$ d $ - 元组$ \ bt $的希尔伯特太空运营商，我们充分确定了广义球形aluthge变换$ \ dbt $的光谱图，我们证明可以根据$ \ dbt $的迭代型规范来计算$ \ bt $的光谱半径。 \ \ $ \ bm {t} \ equiv（t_1，\ cdots，t_d）$是通勤$ d $ - 在无限尺寸可分离的希尔伯特空间上作用的有界操作员的元组，让$ p：= \ sqrt { \ begin {array} {c} t_1 \\ \ \ vdots \\ t_d \ end {array} \ right）=）= \ left（\ begin {array} {c} v_1 \\ \ \ vdots \\ v_d \ end end {array} \ right）p $$是规范的极性分解，带有$（v_1，\ cdots，v_d）$ a（联合）部分等轴测和$$ \ bigcap_ {i = 1}^d \ ker t_i = \ bigCap_ {i = 1} $ \ bm {t} $的广义球形aluthge变换，$$δ_t（\ bm {t}）：=（p^t v_1p^{1-t}，\ cdots，p^t v_dp^{1-t}）。 $$我们还让$ \ left \ | \ bm {t} \ right \ | _2：= \ left \ | p \ right \ | $。 \我们首先在$ \ bm {t} $的频谱图中确定$Δ_T（\ bm {t}）$的光谱图片;特别是，我们证明，对于任何$ 0 \ le t \ le 1 $，$δ_t（\ bm {t}）$和$ \ bm {t} $具有相同的泰勒光谱，相同的泰勒必需谱，相同的弗雷霍尔姆intex和相同的harte spectrum。 \然后我们研究联合光谱半径$ r（\ bm {t}）$，并证明$ r（\ bm {t}）= \ lim_n \ left \ weft \ |δ_t^{（n）}（n）}（\ bm {t}）（\ bm {t}）表示$ n $ - $Δ_T$的迭代。 \对于$ d = t = 1 $，我们举例说明上述公式失败。
For an arbitrary commuting $d$--tuple $\bT$ of Hilbert space operators, we fully determine the spectral picture of the generalized spherical Aluthge transform $\dbT$ and we prove that the spectral radius of $\bT$ can be calculated from the norms of the iterates of $\dbT$. \ Let $\bm{T} \equiv (T_1,\cdots,T_d)$ be a commuting $d$--tuple of bounded operators acting on an infinite dimensional separable Hilbert space, let $P:=\sqrt{T_1^*T_1+\cdots+T_d^*T_d}$, and let $$ \left( \begin{array}{c} T_1 \\ \vdots \\ T_d \end{array} \right) = \left( \begin{array}{c} V_1 \\ \vdots \\ V_d \end{array} \right) P $$ be the canonical polar decomposition, with $(V_1,\cdots,V_d)$ a (joint) partial isometry and $$ \bigcap_{i=1}^d \ker T_i=\bigcap_{i=1}^d \ker V_i=\ker P. $$ \medskip For $0 \le t \le 1$, we define the generalized spherical Aluthge transform of $\bm{T}$ by $$ Δ_t(\bm{T}):=(P^t V_1P^{1-t}, \cdots, P^t V_dP^{1-t}). $$ We also let $\left\|\bm{T}\right\|_2:=\left\|P\right\|$. \ We first determine the spectral picture of $Δ_t(\bm{T})$ in terms of the spectral picture of $\bm{T}$; in particular, we prove that, for any $0 \le t \le 1$, $Δ_t(\bm{T})$ and $\bm{T}$ have the same Taylor spectrum, the same Taylor essential spectrum, the same Fredholm index, and the same Harte spectrum. \ We then study the joint spectral radius $r(\bm{T})$, and prove that $r(\bm{T})=\lim_n\left\|Δ_t^{(n)}(\bm{T})\right\|_2 \,\, (0 < t < 1)$, where $Δ_t^{(n)}$ denotes the $n$--th iterate of $Δ_t$. \ For $d=t=1$, we give an example where the above formula fails.
