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- W3020008093 abstract "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 ${ [x,y,z_1,z_2,t]in {mathbb C}{mathbb P}^4: x^n+y^n+(-1)^{n-1}(e^{2pi I/n}z_1+z_2)^n-t^n=0}$, is generated by the Fourier matrix $F_n$." @default.
- W3020008093 created "2020-05-01" @default.
- W3020008093 creator A5033015591 @default.
- W3020008093 creator A5067477747 @default.
- W3020008093 date "2020-04-22" @default.
- W3020008093 modified "2023-09-26" @default.
- W3020008093 title "Spectral surfaces for operator pairs and Hadamard matrices of F type" @default.
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- W3020008093 doi "https://doi.org/10.48550/arxiv.2004.10903" @default.
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