Linear operators preserving the decomposable numerical range

Canadian Mathematical Bulletin, 2000

Suppose m and n are integers such that 1 m n. For a subgroup H of the symmetric group S m of degree m, consider the generalized matrix function on m m matrices B = (b ij ) de ned by d H (B) = P 2H Q m j=1 b j (j) and the generalized numerical range of an n n complex matrix A associated with d H de ned by W H (A) = fd H (X AX) : X is n m such that X X = I m g:

Linear and Multilinear Algebra, 1996

Let A, C be n × n complex matrices. We prove in the affirmative the conjecture that the C-numerical range of A, defined by W C (A) = {tr (CU * AU) : U is unitary} , is always star-shaped with respect to star-center (tr A)(tr C)/n. This result is equivalent to that the image of the unitary orbit {U * AU : U is unitary} of A under any complex linear functional is always star-shaped.

Numerische Mathematik, 1976

For each norm v on <en, we define a numerical range Z., which is symmetric in the sense that Z. =Z"D, where v D is the dual norm. We prove that, for aE <e nn , Z.(a) contains the classical field of values V(a). In the special case that v is the lcnorm, Z.(a) is contained in a set G(a) of Gershgorin type defined by C. R. Johnson. When a is in the complex linear span of both the Hennitians and the v-Hennitians, then Z.(a), V(a) and the convex hull of the usual v-numerical range V.(a) all coincide. We prove some results concerning points of V(a) which are extreme points of Z.(a).

Linear Algebra and its Applications, 1999

In [18], among other equivalent conditions, it is proved that a square complex matrix A is permutationally similar to a block-shift matrix if and only if for any complex matrix B with the same zero pattern as A, W (B), the numerical range of B, is a circular disk centered at the origin. In this paper, we add a long list of further new equivalent conditions. The corresponding result for the numerical range of a square complex matrix to be invariant under a rotation about the origin through an angle of 2π/m, where m 2 is a given positive integer, is also proved. Many interesting by-products are obtained. In particular, on the numerical range of a square nonnegative matrix A, the following unexpected results are established: (i) when the undirected graph of A is connected, if W (A) is a circular disk centered at the origin, then so is W (B), for any complex matrix B with the same zero pattern as A; (ii) when A is irreducible, if λ is an eigenvalue in the peripheral spectrum of A that lies on the boundary of W (A), then λ is a sharp point of W (A). We also obtain results on the numerical range of an irreducible square nonnegative matrix, which strengthen or clarify the work of Issos [9] and Nylen and Tam [14] on this topic. Open questions are posed at the end.

Linear Algebra and its Applications, 1990

Let F be a surjective linear mapping between the algebras L(H) and L(K) of all bounded operators on nontrivial complex Hilbert spaces H and K respectively. For any positive integer k let W,(A) denote the kth numerical range of an operator A on H. If k is strictly less than one-half the dimension of H and W,(F(A)) = Wk. A) for ah A from L(H), then there is a unitary mapping U: H + K such that either F(A) = UAu* or F(A) = (UAU*)' for every A E L(H), where the transposition is taken in any basis of K, fixed in advance. This generalizes the result of S. Pierce and W. Watkins on finite-dimensional spaces. The case of k greater than or equal to one-half of the dimension of H is also treated using our method. Our proofs depend on a characterization of those linear operators preserving projections of rank one, which is of independent interest.

Electronic Journal of Linear Algebra, 2013

A subset S of the complex plane has n-fold symmetry about the origin (n-sato) if z ∈ S implies e 2π n z ∈ S. The 3 × 3 matrices A for which the numerical range W (A) has 3-sato have been characterized in two ways. First, W (A) has 3-sato if and only if the spectrum of A has 3-sato while tr(A 2 A *) = 0. In addition, W (A) has 3-sato if and only if A is unitarily similar to an element of a certain family of generalized permutation matrices. Here it is shown that for an n × n matrix A, if a specific finite collection of traces of words in A and A * are all zero, then W (A) has n-sato. Further, this condition is shown to be necessary when n = 4. Meanwhile, an example is provided to show that the condition of being unitarily similar to a generalized permutation matrix does not extend in an obvious way.

2013

For a positive integer n, let Mn be the set of n×n complex matrices. Suppose m, n ≥ 2 are positive integers and k ∈ {1, . . . , mn − 1}. Denote by W k (X) the k-numerical range of a matrix X ∈ Mmn. It is shown that a linear map φ : Mmn → Mmn satisfies

Linear Algebra and its Applications, 2010

Generalized matrix functions Let H be a subgroup of the symmetric group of degree m and let χ be an irreducible character of H. In this paper we give conditions that characterize the pairs of matrices that leave invariant the value of a generalized matrix function associated with H and χ , on the set of the upper triangular matrices.

1991

We give equivalent characterizations for those n x n complex matrices A whose unitary orbits %?(A) and C-numerical ranges WC{A) satisfy ei8&(A) = f/(A) or e'e WC(A) = WC(A) for some real 0 (or for all real 0). In particular, we show that they are the block-cyclic or block-shift operators. Some of these results are extended to infinite-dimensional Hubert spaces.

Doklady Mathematics, 2009