Gelfand and Ponomarev [Functional Anal. Appl. 3 (1969) [325][326] proved that the problem of classifying pairs of commuting linear operators contains the problem of classifying k-tuples of linear operators for any k. We prove an analogous statement for semilinear operators.

L. Huang [Consimilarity of quaternion matrices and complex matrices, Linear Algebra Appl. 331 (2001) 21-30] gave a canonical form of a quaternion matrix with respect to consimilarity transformations A ↦S −1 AS in which S is a nonsingular quaternion matrix and h = a + bi + cj + dk ↦h ∶= a − bi + cj − dk (a, b, c, d ∈ R).We give an analogous canonical form of a quaternion matrix with respect to consimilarity transformations A ↦Ŝ −1 AS in which h ↦ĥ is an arbitrary involutive automorphism of the skew field of quaternions. We apply the obtained canonical form to the quaternion matrix equations AX −XB = C and X − AXB = C.

For a given poset, we consider its representations by systems of subspaces of a unitary space ordered by inclusion. We classify such systems for all posets for which an explicit classification is possible.Comment: 20 page

We give a canonical form of matrices of a cycle of linear or semilinear mapping V_1 --- V_2 --- ... --- V_t --- V_1 in which all V_i are complex vector spaces, each line is an arrow ---> or <---, and each arrow denotes a linear or semilinear mapping.Comment: 18 page

We study systems of linear and semilinear mappings considering them as representations of a directed graph G with full and dashed arrows: a representation of G is given by assigning to each vertex a complex vector space, to each full arrow a linear mapping, and to each dashed arrow a semilinear mapping of the corresponding vector spaces. We extend to such representations the classical theorems by Gabriel about quivers of finite type and by Nazarova, Donovan, and Freislich about quivers of tame types.