二次型算子: 數值域 相似與超不變子空間格子

Abstract

一個在希氏空間的算子滿足二次多項方程，我們稱作二次型算子。在本論 文中，我們證明了二次型算子的數值域是橢圓盤或是其退化形，另給了其 為閉集的充要條件。再來我們研究二次型算子之間的關係, 而得到兩個平 方零算子或兩個冪等算子的酉等價、相似及擬相似之充要條件。最後我們 決定了二次型算子的超不變子空間格子。 In this thesis, first we show that the numerical range of a quadratic operator on a Hilbert space must be an elliptical disc or its degenerate form. We also obtain necessary and sufficient conditions for such an operator to attain its numerical radius. We also study the similarity of quadratic operators. We obtain necessary and sufficient conditions the unitary equivalence and similarity of two square zero operators and also two idempotent operators. Finally, we determine the hyperlattices of quadratic operators.