Hou, Bo; Gao, Suogang The structure of some linear transformations. (English) Zbl 1253.15005 Linear Algebra Appl. 437, No. 9, 2110-2116 (2012). 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)]. Reviewer: John D. Dixon (Ottawa) MSC: 15A04 Linear transformations, semilinear transformations 15A18 Eigenvalues, singular values, and eigenvectors 15A63 Quadratic and bilinear forms, inner products Keywords:Hessenberg pairs; tridiagonal pairs; split decomposition; linear transformations; bilinear forms; eigenspaces Citations:Zbl 1146.17009; Zbl 0995.05148 PDFBibTeX XMLCite \textit{B. Hou} and \textit{S. Gao}, Linear Algebra Appl. 437, No. 9, 2110--2116 (2012; Zbl 1253.15005) Full Text: DOI References: [1] Godjali, Ali, Hessenberg pairs of linear transformations, Linear Algebra Appl., 431, 1579-1586 (2009) · Zbl 1176.15004 [2] Godjali, Ali, Thin Hessenberg pairs, Linear Algebra Appl., 432, 3231-3249 (2010) · Zbl 1209.15007 [3] Hou, B.; Gao, S., The shape of linear transformations, Linear Algebra Appl., 433, 2088-2095 (2010) · Zbl 1216.15003 [4] Ito, T.; Tanabe, K.; Terwilliger, P., Some algebra related to \(P\)-and \(Q\)-polynomial association schemes, (Dimacs Ser. Discrete Math. Theoret. Comput. Sci., vol. 56 (2001), American Mathematical Society), 167-192 · Zbl 0995.05148 [5] Ito, T.; Terwilliger, P., \(q\)-inverting pairs of linear transformations and the \(q\)-tetrahedron algebra, Linear Algebra Appl., 426, 516-532 (2007) · Zbl 1146.17009 [6] Nomura, K.; Terwilliger, P., The split decomposition of a tridiagonal pair, Linear Algebra Appl., 424, 339-345 (2007) · Zbl 1127.05108 [7] Nomura, K.; Terwilliger, P., Sharp tridiagonal pairs, Linear Algebra Appl., 429, 79-99 (2008) · Zbl 1193.05164 [8] Nomura, K.; Terwilliger, P., The structure of a tridiagonal pair, Linear Algebra Appl., 429, 1647-1662 (2008) · Zbl 1144.05331 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.