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I wonder if some work has been developed on operators in Hilbert space that have the property of having matrices instead of numbers as eigenvalues (the matrices do not necessarily act on vectors in the Hilbert space - and they can be of a different dimension).

If such a framework akready exists, what are the main properties of these operators ? (References are welcome).

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    $\begingroup$ How do you define "eigenvalue" if you can't multiply a vector by an eigenvalue, as seems to be suggested by saying "the matrices do not necessarily act on vectors"? ${}\qquad{}$ $\endgroup$ Aug 22, 2014 at 1:25
  • $\begingroup$ This isn't about Hilbert spaces but maybe it relates to what you are thinking about. The current thinking on multidimensional determinants at least, is that they should be polynomials in the entries of the matrices (see Gel'fand, I; Kapranov, M.; and Zelevinsky, A., (1992) “Hyperdeterminants”, Advances in Mathematics, 96, pp. 226-263). If you want the determinant to be the product of the eigenvalues of the matrix, then the eigenvalues would have to be scalars. $\endgroup$ Aug 22, 2014 at 3:11
  • $\begingroup$ Here is what I mean. Let $\mathcal H$ be a $d$ dimensional Hilbert space. $\hat M$ is a matrix linear operator on $\mathcal H$ with $n \times p$ eigenmatrix $m$ and eigen vector $v$ if $\hat M v = m v$. $\endgroup$
    – Mathias
    Aug 22, 2014 at 4:03

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