Kazuyoshi Okubo

Uniqueness of matrix square roots and an application

Abstract Let A∈M n ( C ) . Let σ(A) denote the spectrum of A , and F(A) the field of values of A . It is shown that if σ(A)∩(−∞,0]=∅ , then A has a unique square root B∈M n ( C ) with σ(B) in the open right (complex) half plane. This result and Lyapunovs theorem are then applied to prove that if F(A)∩(−∞,0]=∅ , then A has a unique square root with positive definite Hermitian part. We will also answer affirmatively an open question about the existence of a real square root B∈M n ( R ) for A∈M n ( R ) with F(A)∩(−∞,0]=∅ , where the field of values of B is in the open right half plane.

Induced norms of the Schur multiplier operator

Fix an n-by-n complex matrix A, and consider the operator X ↦ SA(X) ≡ A ∘ X on n-by-n complex matrices X, where A∘X denotes the Schur product of A and X. We show that the induced norm of SA with respect to the numerical-radius norm is at most one if and only if the matrix A admits a factorization A=B∗WB, where W is a contractive matrix and the Euclidean norms of the columns of B are at most one. We give other equivalent characterizations and derive, as a consequence, a formally similar result about the induced norm of SA with respect to the spectral norm.

Hölder-type norm inequalities for schur products of matrices

Abstract For a unitarily invariant norm ∥·∥φon M n and p ⩾ 1 we define ∥Aφ, p, by ∥∣A∣p∥ 1 p φ. Then ∥·∥φ, p is again a unitarily invariant norm. We give Holder-type inequalities for Schur products of the form ∥A∘B∥φ0,p0⩽∥A∥φ1,p1·∥B∥φ2,p2.. As a corollary, we settle, in a stronger form, a conjecture of Marcus et al. on submultiplicativity of a unitarily invariant norm with respect to Schur multiplication. We prove also Holder-type inequalities for operator radii wρ(·) of the form wσρ (A∘B) ⩽ wσ(A)· wρ(B).

On weakly unitarily invariant norm and the Aluthge transformation

Abstract In this paper we show that ⦀f(P λ UP 1−λ )⦀⩽ max ⦀f(T)⦀, ⦀U ∗ ·f(T)·U+f(0)(I−U ∗ U)⦀ , where T∈ B ( H ) , ⦀·⦀ is a semi-norm on B ( H ) which satisfies some conditions, T=UP (polar decomposition), 0⩽λ⩽1 and f is a polynomial. As a consequence of this fact, we will show that some semi-norms ⦀·⦀ including the ρ-radii (0 ⦀f(P λ UP 1−λ )⦀⩽⦀f(T)⦀ . We also give some related results.

Uniqueness of matrix square roots under a numerical range condition

For A∈Mn(C), let W(A) denote the numerical range of A. It is shown that if W(A)∩(−∞,0)=∅, then A has a unique square root B∈Mn(C) with W(B) in the closed right half plane.

On generalized numerical range of the Aluthge transformation

Abstract In this paper the authors show that the Aluthge transformation T of a matrix T and a polynomial f satisfy the inclusion relation W C (f( T ))⊂W C (f(T)) for the generalized numerical range if C is a Hermitian matrix or a rank-one matrix.

ϱ-contraction and 2 × 2 matrix

Abstract In this paper, the following is proved. When |a|⩽1 and |b|⩽1, A = a c O b is a g9-contraction if and only if |c| 2 + |a − | 2 ⩽ inf ζ∈D {ρ + (1 −ρ) a ζ }{ρ + (1 − ρ)bζ} − a ζ|ζ| 2 ρζ 2 where D is the open unit disc. The result is then extended to quadratic operators. Several special cases of the result are analysed in detail.

On the characterization of 2×2ρ-contraction matrices

Abstract We give an explicit description of all ρ -contractive (in Nagy–Foias sense) 2×2 matrices. This description leads to the formulas for ρ -numerical radii when the eigenvalues of such matrices either have equal absolute values or equal ( mod π ) arguments. We also discuss (natural) generalizations to the case of decomposable operators with at most two-dimensional blocks covering, in particular, the quadratic operators.

Issues Surrounding Teaching Linear Algebra

Linear Algebra is one of the most important courses in the education of mathematicians, scientists, engineers and economists. DG 3 was organized by The Education Committee of International Linear Algebra Society (ILAS) in order to give mathematicians and mathematics educators the opportunity to discuss several issues on teaching and learning Linear Algebra including motivation, challenging problems, visualization, learning technology, preparation in high school, history of Linear algebra and research topics at different levels. Some of these problems were discussed. Around 50 participants participated in the discussion.