On Indecomposable Normal Matrices in Spaces with Indefinite Scalar Product

Finite dimensional linear spaces (both complex and real) with indefinite scalar product [·, ·] are considered. Upper and lower bounds are given for the size of an indecomposable matrix that is normal with respect to this scalar product in terms of specific functions of v = min{v − , v+}, where v− (v+) is the number of negative (positive) squares of the form [x, x]. All the bounds except for one are proved to be strict. 1 Definitions and notation Consider a complex (real) linear space C (R) with an indefinite scalar product [·, ·]. By definition, the latter is a nondegenerate sesquilinear (bilinear) Hermitian form. If the usual scalar product (·, ·) is fixed, then there exists a nonsingular Hermitian operator H such that [x, y] = (Hx, y) ∀x, y ∈ C (R). If A is a linear operator, then the H-adjoint of A (denoted by A) is defined by the identity [Ax, y] ≡ [x,Ay]. An operator N is called H-normal if NN [∗] = N N . An operator U is called H-unitary if UU [∗] = I, where I is the identity transformation. Let V be a nontrivial subspace of C (R). The subspace V is called neutral if [x, y] = 0 ∀x, y ∈ V . If the conditions x ∈ V and [x, y] = 0 ∀y ∈ V imply x = 0, then V is called nondegenerate. The subspace V [⊥] is defined as the set of all vectors x from C (R) such that [x, y] = 0 ∀y ∈ V . If V is nondegenerate, then V [⊥] is also nondegenerate and V +̇V [⊥] = C (R), where +̇ stands for the direct sum. A linear operator A is called decomposable if there exists a nondegenerate proper subspace V of C (R) such that both V and V [⊥] are invariant under A or (it is the same) if V is invariant both under A and A. Then A is the H-orthogonal sum of A1 = A|V and A2 = A|V [⊥] . If an operator A is not decomposable, it is called indecomposable. By the rank of a space we mean v = min{v−, v+}, where v− (v+) is the number of negative (positive) squares of the form [x, x], i.e., the number of negative (positive) eigenvalues of the operator H . The problem is to find functions f1(·), f2(·) such that f1(v) ≤ n ≤ f2(v) for any indecomposable Hnormal operator acting in a space of dimension n and of rank v and to find out whether these bounds are strict. This problem arises in the classification of indecomposable H-normal matrices [2, 3]. The bounds for the size of an indecomposable H-normal matrix in a complex space are known [2]. In Section 2, we check their strictness. The bounds for matrices in real spaces are considered in Section 3. As in [2] and [3], we denote by Ir the r×r identity matrix, by Dr the r×r matrix with 1’s on the trailing diagonal and zeros elsewhere, and by A ⊕ B ⊕ . . .⊕ Z the block diagonal matrix with blocks A, B, . . ., Z. By A we mean A transposed.