Kuo-Zhong Wang

Product of operators and numerical range

We show that a bounded linear operator is a multiple of a unitary operator if and only if and always have the same numerical radius or the same numerical range for all (rank one) . More generally, for any bounded linear operators , we show that and always have the same numerical radius (resp., the same numerical range) for all (rank one) if and only if (resp., ) is a multiple of a unitary operator for some . We extend the result to other types of generalized numerical ranges including the -numerical range and the higher rank numerical range.

Numerical radii for tensor products of matrices

Abstract For-by- and-by- complex matrices and, it is known that the inequality holds, where and denote, respectively, the numerical radius and the operator norm of a matrix. In this paper, we consider when this becomes an equality. We show that (1) if and, then one of the following two conditions holds: (i) has a unitary part, and (ii) is completely nonunitary and the numerical range of is a circular disc centered at the origin, (2) if for some , , then , and, moreover, the equality holds if and only if is unitarily similar to the direct sum of the -by- Jordan block and a matrix with , and (3) if is a nonnegative matrix with its real part (permutationally) irreducible, then , if and only if either or and is permutationally similar to a block-shift matrixwith , where and .

Operator norms and lower bounds of generalized Hausdorff matrices

Let A = (a n,k ) n,k≥0 be a non-negative matrix. Denote by L p,q (A) the supremum of those L satisfying the following inequality: The purpose of this article is to establish a Bennett-type formula for and a Hardy-type formula for and , where is a generalized Hausdorff matrix and 0 < p ≤ 1. Similar results are also established for and for other ranges of p and q. Our results extend [Chen and Wang, Lower bounds of Copson type for Hausdorff matrices, Linear Algebra Appl. 422 (2007), pp. 208–217] and [Chen and Wang, Lower bounds of Copson type for Hausdorff matrices: II, Linear Algebra Appl. 422 (2007) pp. 563–573] from to with α ≥ 0 and completely solve the value problem of , , and for α ∈ ℕ ∪ {0}.

Equality of higher-rank numerical ranges of matrices

Let denote the rank- numerical range of an -by- complex matrix . We give a characterization for , where , via the compressions and the principal submatrices of . As an application, the matrix satisfying , where is the classical numerical range of and , is under consideration. We show that if for some , then is unitarily similar to , where is a 2-by-2 matrix, is a -by- matrix and .

Lower bounds of Copson type for weighted mean matrices and Nörlund matrices

Let 1 ≤ p ≤ ∞, 0 < q ≤ p, and A = (a n,k ) n,k≥0 ≥ 0. Denote by L p,q (A) the supremum of those L satisfying the following inequality: whenever and X ≥ 0. In this article, the value distribution of L p,q (A) is determined for weighted mean matrices, Nörlund matrices and their transposes. We express the exact value of L p,q (A) in terms of the associated weight sequence. For Nörlund matrices and some kinds of transposes, this reduces to a quotient of the norms of such a weight sequence. Our results generalize the work of Bennett.

Circular numerical ranges of partial isometries

Let be an -by- partial isometry whose numerical range is a circular disc with centre and radius . In this paper, we are concerned with the possible values of and . We show that must be if is at most and conjecture that the same is true for the general . As for the radius, we show that if , then the set of all possible values of is . Indeed, it is shown more precisely that for , , the possible values of are those in the interval . In the proof process, we also characterize -by- partial isometries which are (unitarily) irreducible. The paper is concluded with a question on the rotational invariance of nilpotent partial isometries with circular numerical ranges centred at the origin.