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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
@@ Line 37: / Line 37: @@
      * 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.
+===== 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 )].

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