论文标题
F型的操作员对和Hadamard矩阵的光谱表面
Spectral surfaces for operator pairs and Hadamard matrices of F type
论文作者
论文摘要
众所周知，一般而言，在操作员元素的投射关节频谱中，有限多样性的代数性超浮肿的外观并不意味着存在有限维的常见不变子空间。 {\ Mathbb C} {\ Mathbb P}^3：X^n+y^n+（-1）^{n-1} Z^n-T^n = 0 \} $，在某些温和条件下，这意味着存在$ n $ n $ n $ n $ n $不变的子空间的存在。 ITBIS表明，该表面的外观与复杂的Hadamard矩阵有关系。我们为f类型的hadamard矩阵提供了足够的条件，可以生成这种对$ a，b $。对于尺寸$ n = 3,4,5 $，其中有comp [lex hadamard矩阵的完整描述，这也是必要的。最后，我们证明了一对$ a，b $，使得$ a，b，ab $和$ ba $的投影关节频谱包含$ \ {[x，y，z_1，z_1，z_2，t] \ in {MathBb c} {\ Mathbb P}^4： x^n+y^n+（-1）^{n-1}（e^{2πi/n} z_1+z_2）^n-t^n = 0 \} $，由傅立叶矩阵$ f_n $生成。
It is well-known that, in general, an appearance of an algebraic hypersurface of finite multiplicity in the projective joint spectrum of an operator tuple does not imply the existence of a finite-dimensional common invariant subspace.We prove that if for a pair of operators A,B the project time joint spectrum of $A, B$ and $AB$ contains the surface $\{[x,y,z,t]\in {\mathbb C}{\mathbb P}^3: x^n+y^n+(-1)^{n-1}z^n-t^n=0\}$, the under some mild conditions this implies the existence of a subspace of dimension $n$ invariant for both $A$ and $B$. Itbis shown that the appearance of this surface has a relation to complex Hadamard matrices. We give a sufficient condition for a Hadamard matrix of F type to generate such pair $A,B$. For dimensions $n=3,4,5$ where there is a complete description of comp[lex Hadamard matrices, this condition proved to be necessary as well. Finally, we prove that a pair $A,B$ such that the projective joint spectrum of $A,B,AB$ and $BA$ contains $\{ [x,y,z_1,z_2,t]\in {mathbb C}{\mathbb P}^4: x^n+y^n+(-1)^{n-1}(e^{2πI/n}z_1+z_2)^n-t^n=0\}$, is generated by the Fourier matrix $F_n$.
