算子和矩陣的數值域研究

在本論文中, 我們對於某些算子和矩陣的數值域做一個研究。首先, 我們考慮 $C_0$ 收縮算子和二次算子。我們證明如果 $A$ 是一個 $C_0$ 收縮算子有著最小函數 $phi$ 使得 $w(A)=w(S(phi))$ 而且 $B$ 和$ A$ 可交換, 其中 $w(cdot)$ 表示算子的數值半徑, 則 $w(AB)leq w(A)|B|$。因此, 對於所有二次算子 $A$ 和任意和 $A$ 可以交換的算子 $B$, 我們也得到 $w(AB)leq w(A)|B|$。接著, 令 $A$ 代表 $n$ 維($ngeq 2$)的帶權移動矩陣 $[t_{ij}]_{i,j=1}^{n}$, 其中 $t_{i,i+1}=a_i$, $i=1, 2, ldots, n-1$, $t_{n,1}=a_n$ 而且其餘 $t_{i,j}=0$。我們證明它的數值域邊界具有一個線段的充分必要條件是這些 $a_i$ 不為零而且 $A$ 的 $n-1$ 維主要子矩陣的數值域都相同。由此我們得到如果一個 $n$ 維的帶權移動矩陣 $A$, 其中這些 $a_i$ 是非零而且 $|a_i|$ 是週期的, 則它的數值域邊界具有一個線段。我們也證明了它的數值域邊界含有一個非圓的橢圓弧若且唯若這些 $a_i$ 不為零, $n$ 是偶數, $|a_1|=|a_3|=cdots=|a_{n-1}|$, $|a_2|=|a_4|=cdots=|a_n|$ 而且 $|a_1| eq |a_2|$。最後, 我們刻劃 $A$ 是可約的情形而且完整描述它的數值域。再來, 我們證明一個四維的實冪零矩陣 $A$ 的數值域邊界最多具有兩個線段。我們也給了一個四維的冪零矩陣 $A$ 的數值域邊界具有兩個平行線段的一個充分必要條件。最後，我們將證明一個有限維矩陣 $A=(sum_{i=1}^{k_1}oplus A_i)oplus{dia (w_1, ldots, w_{k_2})}$, 其中 $A_i=left[egin{array}{cc}x_i & z_i\ 0 & y_iend{array} ight], i=1, ldots,k_1$, 是兩個非負收縮矩陣的乘積若且唯若 $0leq x_i, y_i, w_jleq 1$ 而且, 對於所有 $i, j$, $|z_i|leq |sqrt{x_i}-sqrt{y_i}|sqrt{(1-x_i)(1-y_i)}$。藉此我們可以在 $n$ 維的二次算子上得到一個類似的結果。

關鍵字

數值域； C0 收縮算子；二次算子；帶權移動矩陣

並列摘要

In this thesis, we study properties of the numerical ranges of some operators and matrices. First, we consider $C_0$ contractions and quadratic operators. We show that if $A$ is a $C_0$ contraction with minimal function $phi$ such that $w(A)=w(S(phi))$ and if $B$ commutes with $A$, where $w(cdot)$ denotes the numerical radius of an operator, then $w(AB)leq w(A)|B|$. As a consequence, we also obtain $w(AB)leq w(A)|B|$ for any quadratic operator $A$ and any $B$ commuting with $A$. Second, let $A$ be the $n$-by-$n$ ($ngeq 2$) weighted shift matrix $[t_{ij}]_{i,j=1}^{n}$, where $t_{i,i+1}=a_i$ for $i=1,2,cdots,n-1$, $t_{n,1}=a_n$ and $t_{i,j}=0$ otherwise. We show that the boundary of its numerical range contains a line segment if and only if the $a_i$'s are nonzero and the numerical ranges of the ($n-1$)-by-($n-1$) principal submatrices of $A$ are all equal. Using this, we obtain that the boundary of the numerical range of an $n$-by-$n$ weighted shift matrix $A$ has a line segment if the $a_i$'s are nonzero and their moduli are periodic. We also prove that $partial W(A)$ contains a noncircular elliptic arc if and only if the $a_i$'s are nonzero, $n$ is even, $|a_1|=|a_3|=cdots=|a_{n-1}|$, $|a_2|=|a_4|=cdots=|a_n|$ and $|a_1| eq |a_2|$. Finally, we give a criterion for $A$ to be reducible and completely characterize the numerical ranges of such matrices. Next, we show that if $A$ is a $4$-by-$4$ nilpotent real matrix, then the boundary of its numerical range has at most two line segments. We also give a necessary and sufficient condition for the boundary of $W(A)$ to have a pair of parallel line segments. Finally, we give a necessary and sufficient condition for a finite matrix $A=(sum_{i=1}^{k_1}oplus A_i)oplus{dia (w_1, ldots, w_{k_2})}$, where $A_i=left[egin{array}{cc}x_i & z_i\ 0 & y_iend{array} ight]$ for all $i$, to be a product of two nonnegative contractions: $0leq x_i, y_i, w_jleq 1$ and $|z_i|leq |sqrt{x_i}-sqrt{y_i}|sqrt{(1-x_i)(1-y_i)}$ for all $i$, $j$. Applying this, we obtain an analogous characterization for an $n$-by-$n$ quadratic operator.

並列關鍵字

numerical range ； C0 contraction ； quadratic operator ； weighted shift matrix

參考文獻

[1] H. Bercovici, Operator Theory and Arithmetic in $H^{infty}$, Amer. Math. Soc., Providence, 1988.

[3] C. A. Berger, J. G. Stampfli, Mapping theorems for the numerical range, Amer. J. Math. 89 (1967) 1047--1055.

[4] R. Bouldin, The numerical range of a product, II, J. Math. Anal. Appl. 33 (1971) 212--219.

[5] M.-T. Chien, H. Nakazato, The numerical radius of a weighted shift operator with geometric weights, Electron. J. Linear Algebra 18 (2009) 58--63.

[6] M.-T. Chien, H. Nakazato, Flat portions on the boundary of the numerical ranges of certain Toeplitz matrices, Linear Multilinear Algebra 56 (2008) 143--162.

國際替代計量

算子和矩陣的數值域研究

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