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numerical-range:generalizations:c-numerical-range [2019/03/01 16:45]
plewandowska [Properties]
numerical-range:generalizations:c-numerical-range [2020/05/16 10:42] (current)
plewandowska
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      * there exist $\mu \in \mathbb{C}$ such that $C - \mu \1$ is unitarity similar to $(C_{ij})_{1 \le i,j \le d}$ in block form, where $C_{ii}$ are square matrices and $C_{ij} = 0$ if $ i \neq j+1$;      * there exist $\mu \in \mathbb{C}$ such that $C - \mu \1$ is unitarity similar to $(C_{ij})_{1 \le i,j \le d}$ in block form, where $C_{ii}$ are square matrices and $C_{ij} = 0$ if $ i \neq j+1$;
      * there exist $\mu \in \mathbb{C}$ such that $C - \mu \1$ has rank one.       * there exist $\mu \in \mathbb{C}$ such that $C - \mu \1$ has rank one. 
 +
 +===== Generalization =====
 +We can generalize the $C$-numerical range $W_C(A)$ ​ to Schatten-class operators i.e. to
 +$C\in\mathcal B^p(\mathcal H)$ and $A\in\mathcal B^q(\mathcal H)$ with condition $1/p + 1/q = 1$, and show that
 +its closure is always star-shaped with respect to the origin [( :zhang2019c )].
 +
 +
 +
 +Let $\mathcal{X},​ \mathcal{Y}$ denote an arbitrary infinite-dimensional separable complex Hilbert space. Moreover, $\mathcal B(\mathcal X,\mathcal Y)$, $\mathcal K(\mathcal X,\mathcal Y)$ and 
 +$\mathcal B^p(\mathcal X,\mathcal Y)$ denote the set of all bounded, compact and $p$-th Schatten-class operators between $\mathcal X$ and $\mathcal Y$, respectively. ​ By $\mathcal B^p( \mathcal{X},​ \mathcal{Y})$ we denote all 
 +$p$-Schatten-class operators defined by
 +
 +\begin{equation}
 +\mathcal B^p(\mathcal X,\mathcal Y)
 +:= \Big\lbrace C \in\mathcal K(\mathcal X,\mathcal Y)\,​\Big|\,​\sum\nolimits_{n=1}^\infty s_n(C)^p<​\infty\Big\rbrace
 +\end{equation}
 +for $p\in [1,\infty)$ whereas the Schatten-$p$-norm of matrix $A$
 +\begin{equation}
 +||A||_p := \Big(\sum_{n=1}^\infty s_n(A)^p\Big)^{1/​p}
 +\end{equation}
 +where sequence $(s_n)_{n=1}^{\infty}$ comes from above well-know Schmidt decomposition theorem.
 +
 +
 +
 +
 +
 +===Schmidt decomposition===
 +For each $C \in \mathcal K(\mathcal X,\mathcal Y)$, there exists a decreasing null sequence ​
 +$(s_n(C))_{n\in\mathbb N}$ in $[0,​\infty)$ as well as orthonormal systems $(f_n)_{n\in\mathbb N}$ in $\mathcal X$
 +and $(g_n)_{n\in\mathbb N}$ in $\mathcal Y$ such that
 +\begin{align*}
 +C = \sum_{n=1}^\infty s_n(C)\langle f_n,​\cdot\rangle g_n\,,
 +\end{align*}
 +where the series converges in the operator norm.
 +
 +Moreover, the sequence $(s_n(C))_{n\in\mathbb N}$ is uniquely determined by $C$.
 +
 +
 +
 +===Definition===
 +Let $p,q\in [1,\infty]$ be conjugate. Then for $C\in\mathcal B^p(\mathcal H)$
 +and $A\in\mathcal B^q(\mathcal H)$, we define the \emph{$C$-numerical range} of $T$ to be
 +\begin{equation}
 +W_C (A):​=\lbrace \operatorname{tr}(CU^\dagger AU)\,​|\,​U\in\mathcal B(\mathcal H)\text{ unitary}\rbrace\,​.
 +\end{equation}
 +
 +
 +The properties of $C$-numerical range in infinite-dimensional vector space for Schatten-class operator we can find in [( :dirr2018c )].
 +
  
numerical-range/generalizations/c-numerical-range.1551458718.txt.gz · Last modified: 2019/03/01 16:45 by plewandowska