Spectral hypersurfaces for operator pairs and Hadamard matrices of F type

Abstract

We prove that if for a pair of $$n\times n$$ matrices A and B, with A being normal, the projective joint spectrum of A, B, $$\mathrm{AB}$$ and the identity is given by $$\begin{aligned} \sigma (A,B,\mathrm{AB},I)=\{[x,y,z,t]\in {\mathbb {C}}{\mathbb P}^3:x^n+y^n+(-1)^{n-1}z^n-t^n=0 \}, \end{aligned}$$ then this pair is unitary equivalent to a one associated with a complex Hadamard matrix of order n. If $$n=3,4$$ , or 5, where there is a complete description of Hadamard matrices, we list those that generate a pair with the above mentioned spectrum. If $$\begin{aligned}&\sigma (A,B,\mathrm{AB},\mathrm{BA},I) =\{[x,y,z_1,z_2,t]\in {\mathbb {C}}\mathbb { P}^4: \\&x^n+y^n+(-1)^{n-1}(e^{2\pi i/n}z_1+z_2)^n-t^n=0\}, \end{aligned}$$ this Hadamard matrix is exactly the Fourier matrix $$F_n$$ . If for an operator pair A, B acting on a Hilbert space, such hypersurfaces appear as components of the projective joint spectrum of the corresponding tuples, then under some mild conditions the pair has a common invariant subspace of dimension n, and the restriction of A, B to this subspace is generated by a Hadamard matrix of F type.