Spectral hypersurfaces for operator pairs and Hadamard matrices of F type
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Tusi Mathematical Research Group
ORIGINAL PAPER
Spectral hypersurfaces for operator pairs and Hadamard matrices of F type Thomas Peebles1 • Michael Stessin1 Received: 2 July 2020 / Accepted: 12 September 2020 Ó Tusi Mathematical Research Group (TMRG) 2020
Abstract We prove that if for a pair of n n matrices A and B, with A being normal, the projective joint spectrum of A, B, AB and the identity is given by rðA; B; AB; IÞ ¼ f½x; y; z; t 2 CP3 : xn þ yn þ ð1Þn1 zn tn ¼ 0g; then this pair is unitary equivalent to a one associated with a complex Hadamard matrix of order n. If n ¼ 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 rðA; B; AB; BA; IÞ ¼ f½x; y; z1 ; z2 ; t 2 CP4 : xn þ yn þ ð1Þn1 ðe2pi=n z1 þ z2 Þn tn ¼ 0g; this Hadamard matrix is exactly the Fourier matrix Fn . 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. Keywords Projective joint spectrum Complex Hadamard matrix Fourier type Hadamard matrix
Mathematics Subject Classification 47A25 47A13 47A75 47A15 14J70
Communicated by Vladimir Bolotnikov. & Michael Stessin [email protected] Thomas Peebles [email protected] 1
Department of Mathematics and Statistics, University at Albany, Albany, NY 12222, USA
T. Peebles and M. Stessin
1 Introduction and statement of main results A matrix h 2 Mn ðCÞ is called a complex Hadamard matrix, if each entry of h is a unimodular complex number, and rows of the matrix are mutually orthogonal. Of course, this is equivalent to p1ffiffin h being unitary and all entries having the same absolute value. According to [3] these matrices were originally introduced by Sylvester [32] as real matrices with entries 1 and orthogonal rows. Hadamard matrices with entries being roots of unity, denoted by H(q, N) type, where q is the order of the root of unity, and N is the size of the matrix, were introduced by Butson [5]. Matrices with arbitrary complex unimodular entries appeared in [29]. Interest to Hadamard matrices is caused by the fact that they appear in a number of mathematical objects. Among them are: maximal abelian -sublagebras of the algebra of complex n n matrices [29], statistical mechanical model, knot invariants, and planar algebras, [22], to name a few. They are also associated with quantum permutation groups (cf. [1–3]). A list of known families of Hadamard matrices can be found in [35]. As we will see below, complex Hadamard matrices naturally appear in multivariable spectral theory in connection with projective joint spectra of operator pairs. This connection allows us to establish a spectral characterization of Fourier type Hadamard matrices associated with a certain
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