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The structure of some linear transformations. (English) Zbl 1253.15005

Let \(V\) be a finite dimensional vector space over an algebraically closed field. Let \(A_{+}\) and \(A_{-}\) be operators on \(V\) having the same eigenspaces \(V_{0},V_{1},\dots ,V_{d}\), and let \(A_{+}^{\ast}\) and \(A_{-}^{\ast}\) be operators having the same eigenspaces \(V_{0}^{\ast},V_{1}^{\ast} ,\dots ,V_{d}^{\ast}\) (the star does not represent conjugates). Suppose further that for each \(i\leq d\) we have \(A_{+}V_{i}^{\ast}\subseteq V_{0}^{\ast }+\dots+V_{i+1}^{\ast}\) and \(A_{-}V_{i}^{\ast}\subseteq V_{i-1}^{\ast} +\dots+V_{d}^{\ast}\), and that dual conditions hold for \(A_{+}^{\ast}\) and \(A_{-}^{\ast}\). Furthermore assume that there is no subspace of \(V\) different from \(0\) and \(V\) which is invariant under all of the four operators. The problem of investigating the structure of such linear operators was proposed by T. Ito and P. Terwilliger [ibid. 426, No. 2–3, 516–532 (2007; Zbl 1146.17009)].
In the present paper, the authors show that each of the four operators is determined up to an affine transformation by the sequences of eigenspaces. They also provide new criteria for both \((A_{+},A_{+}^{\ast})\) and \((A_{-},A_{-}^{\ast })\) to be tridiagonal pairs, a special case of the conditions above, see [T. Ito, K. Tanabe and P. Terwilliger, Discrete Math. Theor. Comput. Sci. 56, 167–192 (2001; Zbl 0995.05148)].

MSC:

15A04 Linear transformations, semilinear transformations
15A18 Eigenvalues, singular values, and eigenvectors
15A63 Quadratic and bilinear forms, inner products
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References:

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